#set document(title: "6.6 Divide Polynomials", author: "OpenStax / XYZ Homework")
#set page(width: 8.5in, height: auto, margin: 1in)
#import "@preview/cetz:0.5.2"
#set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en")
#show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math"))
#set par(justify: true, leading: 0.62em, spacing: 0.9em)
#set enum(spacing: 1.1em)   // room between list items so tall inline fractions don't collide
#set list(spacing: 1.1em)
#set table(stroke: 0.5pt + rgb("#c7ccd3"))

#let BLUE = rgb("#183B6F")     // brand navy — section bars + example/solution labels (white on navy 11.09:1)
#let ORANGE = rgb("#A94509")   // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text)
#let RED = rgb("#DC2626")      // brand error-600
#let GREEN = rgb("#059669")    // brand success-600 (decoration only; small green text uses green-text #007942)

#show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt,
  fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt,
  text(fill: white, weight: "bold", size: 19pt, it.body))
#show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt,
  fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt,
  text(fill: white, weight: "bold", size: 12pt, it.body))
#show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body)
#show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body)

#let examplebox(label, title, body) = block(width: 100%, breakable: true,
  fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt,
  above: 12pt, below: 12pt)[
  #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt,
    text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]]
  #body]

// rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade
#let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint,
    stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt),
    above: 11pt, below: 11pt)[
    #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)]
    #linebreak() #body]

#let solutionbox(body) = block(above: 4pt, below: 8pt)[
  #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body]

#let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"),
  stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[
  #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))]

// Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a
// consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the
// print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a
// degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero.
#let _STD_W = 3.5
#let _WIDE_W = 5.6
#let _MAX_H = 3.4
#let _ASPECT_WIDE = 2.2
#let _UPSCALE_MAX = 1.15
#let stdfig(body) = context {
  let m = measure(body)
  let w = calc.max(m.width / 1in, 0.01)
  let h = calc.max(m.height / 1in, 0.01)
  let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W }
  let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX)
  align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body)))
}

#show figure: set block(breakable: false)
#set figure(gap: 8pt)
#show figure.caption: set text(size: 8.5pt, fill: rgb("#555"))


== 6.6#h(0.6em)Divide Polynomials

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Add: #math.equation(block: false, alt: "the fraction 3 over d plus the fraction x over d .")[$frac(3, d) + frac(x, d) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 plus x over d")[$frac(3 + x, d)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 30 x y cubed over 5 x y .")[$frac(30 x y^(3), 5 x y) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "6 y squared")[$6 y^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Combine like terms: #math.equation(block: false, alt: "8 a squared plus 12 a plus 1 plus 3 a squared minus 5 a plus 4 .")[$8 a^(2) + 12 a + 1 + 3 a^(2) − 5 a + 4 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "11 a squared plus 7 a plus 5")[$11 a^(2) + 7 a + 5$]
]
]


=== Divide a Polynomial by a Monomial

In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a #strong[monomial].

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The sum,], [#math.equation(block: false, alt: "the fraction y over 5 plus the fraction 2 over 5 ,")[$frac(y, 5) + frac(2, 5) ,$]]),
  [simplifies to], [#math.equation(block: false, alt: "the fraction y plus 2 over 5 .")[$frac(y + 2, 5) .$]],
))

Now we will do this in reverse to split a single fraction into separate fractions.

We’ll state the fraction addition property here just as you learned it and in reverse.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fraction Addition]

If #math.equation(block: false, alt: "a , b , and c")[$a , b , #h(0.2em) "and" #h(0.2em) c$] are numbers where #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], then

#math.equation(block: true, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c and the fraction a plus b over c equals the fraction a over c plus the fraction b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c) #h(1em) "and" #h(1em) frac(a + b, c) = frac(a, c) + frac(b, c)$]
]

We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([For example,], [#math.equation(block: false, alt: "the fraction y plus 2 over 5")[$frac(y + 2, 5)$]]),
  [can be written], [#math.equation(block: false, alt: "the fraction y over 5 plus the fraction 2 over 5 .")[$frac(y, 5) + frac(2, 5) .$]],
))

We use this form of fraction addition to divide polynomials by monomials.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division of a Polynomial by a Monomial]

To divide a polynomial by a monomial, divide each term of the polynomial by the #strong[monomial].
]

#examplebox("Example 1")[][
Find the quotient: #math.equation(block: false, alt: "the fraction 7 y squared plus 21 over 7 .")[$frac(7 y^(2) + 21, 7) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 7 y squared plus 21 over 7")[$frac(7 y^(2) + 21, 7)$]]),
  [Divide each term of the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction 7 y squared over 7 plus the fraction 21 over 7")[$frac(7 y^(2), 7) + frac(21, 7)$]],
  [Simplify each fraction.], [#math.equation(block: false, alt: "y squared plus 3")[$y^(2) + 3$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 8 z squared plus 24 over 4 .")[$frac(8 z^(2) + 24, 4) .$]

#solutionbox[
#math.equation(block: true, alt: "2 z squared plus 6")[$2 z^(2) + 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 18 z squared minus 27 over 9 .")[$frac(18 z^(2) − 27, 9) .$]

#solutionbox[
#math.equation(block: true, alt: "2 z squared minus 3")[$2 z^(2) − 3$]
]
]

Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.

#examplebox("Example 2")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis 18 x cubed minus 36 x squared close parenthesis divided by 6 x .")[$( 18 x^(3) − 36 x^(2) ) ÷ 6 x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 18 x cubed minus 36 x squared close parenthesis divided by 6 x")[$( 18 x^(3) − 36 x^(2) ) ÷ 6 x$]]),
  [Rewrite as a fraction.], [#math.equation(block: false, alt: "the fraction 18 x cubed minus 36 x squared over 6 x")[$frac(18 x^(3) − 36 x^(2), 6 x)$]],
  [Divide each term of the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction 18 x cubed over 6 x minus the fraction 36 x squared over 6 x")[$frac(18 x^(3), 6 x) − frac(36 x^(2), 6 x)$]],
  [Simplify.], [#math.equation(block: false, alt: "3 x squared minus 6 x")[$3 x^(2) − 6 x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 27 b cubed minus 33 b squared close parenthesis divided by 3 b .")[$( 27 b^(3) − 33 b^(2) ) ÷ 3 b .$]

#solutionbox[
#math.equation(block: true, alt: "9 b squared minus 11 b")[$9 b^(2) − 11 b$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 25 y cubed minus 55 y squared close parenthesis divided by 5 y .")[$( 25 y^(3) − 55 y^(2) ) ÷ 5 y .$]

#solutionbox[
#math.equation(block: true, alt: "5 y squared minus 11 y")[$5 y^(2) − 11 y$]
]
]

When we divide by a negative, we must be extra careful with the signs.

#examplebox("Example 3")[][
Find the quotient: #math.equation(block: false, alt: "the fraction 12 d squared minus 16 d over −4 .")[$frac(12 d^(2) − 16 d, −4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 12 d squared minus 16 d over −4")[$frac(12 d^(2) − 16 d, −4)$]]),
  [Divide each term of the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction 12 d squared over −4 minus the fraction 16 d over −4")[$frac(12 d^(2), −4) − frac(16 d, −4)$]],
  [Simplify. Remember, subtracting a negative is like adding a positive!], [#math.equation(block: false, alt: "−3 d squared plus 4 d")[$−3 d^(2) + 4 d$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 25 y squared minus 15 y over −5 .")[$frac(25 y^(2) − 15 y, −5) .$]

#solutionbox[
#math.equation(block: true, alt: "−5 y squared plus 3 y")[$−5 y^(2) + 3 y$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 42 b squared minus 18 b over −6 .")[$frac(42 b^(2) − 18 b, −6) .$]

#solutionbox[
#math.equation(block: true, alt: "−7 b squared plus 3 b")[$−7 b^(2) + 3 b$]
]
]

#examplebox("Example 4")[][
Find the quotient: #math.equation(block: false, alt: "the fraction 105 y to the power 5 plus 75 y cubed over 5 y squared .")[$frac(105 y^(5) + 75 y^(3), 5 y^(2)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 105 y to the power 5 plus 75 y cubed over 5 y squared")[$frac(105 y^(5) + 75 y^(3), 5 y^(2))$]]),
  [Separate the terms.], [#math.equation(block: false, alt: "the fraction 105 y to the power 5 over 5 y squared plus the fraction 75 y cubed over 5 y squared")[$frac(105 y^(5), 5 y^(2)) + frac(75 y^(3), 5 y^(2))$]],
  [Simplify.], [#math.equation(block: false, alt: "21 y cubed plus 15 y")[$21 y^(3) + 15 y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 60 d to the power 7 plus 24 d to the power 5 over 4 d cubed .")[$frac(60 d^(7) + 24 d^(5), 4 d^(3)) .$]

#solutionbox[
#math.equation(block: true, alt: "15 d to the power 4 plus 6 d squared")[$15 d^(4) + 6 d^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 216 p to the power 7 minus 48 p to the power 5 over 6 p cubed .")[$frac(216 p^(7) − 48 p^(5), 6 p^(3)) .$]

#solutionbox[
#math.equation(block: true, alt: "36 p to the power 4 minus 8 p squared")[$36 p^(4) − 8 p^(2)$]
]
]

#examplebox("Example 5")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis 15 x cubed y minus 35 x y squared close parenthesis divided by open parenthesis −5 x y close parenthesis .")[$( 15 x^(3) y − 35 x y^(2) ) ÷ ( −5 x y ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 15 x cubed y minus 35 x y squared close parenthesis divided by open parenthesis −5 x y close parenthesis")[$( 15 x^(3) y − 35 x y^(2) ) ÷ ( −5 x y )$]]),
  [Rewrite as a fraction.], [#math.equation(block: false, alt: "the fraction 15 x cubed y minus 35 x y squared over −5 x y")[$frac(15 x^(3) y − 35 x y^(2), −5 x y)$]],
  [Separate the terms.], [#math.equation(block: false, alt: "the fraction 15 x cubed y over −5 x y minus the fraction 35 x y squared over −5 x y")[$frac(15 x^(3) y, −5 x y) − frac(35 x y^(2), −5 x y)$]],
  [Simplify.], [#math.equation(block: false, alt: "−3 x squared plus 7 y")[$−3 x^(2) + 7 y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 32 a squared b minus 16 a b squared close parenthesis divided by open parenthesis −8 a b close parenthesis .")[$( 32 a^(2) b − 16 a b^(2) ) ÷ ( −8 a b ) .$]

#solutionbox[
#math.equation(block: true, alt: "−4 a plus 2 b")[$−4 a + 2 b$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis −48 a to the power 8 b to the power 4 minus 36 a to the power 6 b to the power 5 close parenthesis divided by open parenthesis −6 a cubed b cubed close parenthesis .")[$( −48 a^(8) b^(4) − 36 a^(6) b^(5) ) ÷ ( −6 a^(3) b^(3) ) .$]

#solutionbox[
#math.equation(block: true, alt: "8 a to the power 5 b plus 6 a cubed b squared")[$8 a^(5) b + 6 a^(3) b^(2)$]
]
]

#examplebox("Example 6")[][
Find the quotient: #math.equation(block: false, alt: "the fraction 36 x cubed y squared plus 27 x squared y squared minus 9 x squared y cubed over 9 x squared y .")[$frac(36 x^(3) y^(2) + 27 x^(2) y^(2) − 9 x^(2) y^(3), 9 x^(2) y) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 36 x cubed y squared plus 27 x squared y squared minus 9 x squared y cubed over 9 x squared y")[$frac(36 x^(3) y^(2) + 27 x^(2) y^(2) − 9 x^(2) y^(3), 9 x^(2) y)$]]),
  [Separate the terms.], [#math.equation(block: false, alt: "the fraction 36 x cubed y squared over 9 x squared y plus the fraction 27 x squared y squared over 9 x squared y minus the fraction 9 x squared y cubed over 9 x squared y")[$frac(36 x^(3) y^(2), 9 x^(2) y) + frac(27 x^(2) y^(2), 9 x^(2) y) − frac(9 x^(2) y^(3), 9 x^(2) y)$]],
  [Simplify.], [#math.equation(block: false, alt: "4 x y plus 3 y minus y squared")[$4 x y + 3 y − y^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 40 x cubed y squared plus 24 x squared y squared minus 16 x squared y cubed over 8 x squared y .")[$frac(40 x^(3) y^(2) + 24 x^(2) y^(2) − 16 x^(2) y^(3), 8 x^(2) y) .$]

#solutionbox[
#math.equation(block: true, alt: "5 x y plus 3 y minus 2 y squared")[$5 x y + 3 y − 2 y^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 35 a to the power 4 b squared plus 14 a to the power 4 b cubed minus 42 a squared b to the power 4 over 7 a squared b squared .")[$frac(35 a^(4) b^(2) + 14 a^(4) b^(3) − 42 a^(2) b^(4), 7 a^(2) b^(2)) .$]

#solutionbox[
#math.equation(block: true, alt: "5 a squared plus 2 a squared b minus 6 b squared")[$5 a^(2) + 2 a^(2) b − 6 b^(2)$]
]
]

#examplebox("Example 7")[][
Find the quotient: #math.equation(block: false, alt: "the fraction 10 x squared plus 5 x minus 20 over 5 x .")[$frac(10 x^(2) + 5 x − 20, 5 x) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 10 x squared plus 5 x minus 20 over 5 x")[$frac(10 x^(2) + 5 x − 20, 5 x)$]]),
  [Separate the terms.], [#math.equation(block: false, alt: "the fraction 10 x squared over 5 x plus the fraction 5 x over 5 x minus the fraction 20 over 5 x")[$frac(10 x^(2), 5 x) + frac(5 x, 5 x) − frac(20, 5 x)$]],
  [Simplify.], [#math.equation(block: false, alt: "2 x plus 1 minus the fraction 4 over x")[$2 x + 1 − frac(4, x)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 18 c squared plus 6 c minus 9 over 6 c .")[$frac(18 c^(2) + 6 c − 9, 6 c) .$]

#solutionbox[
#math.equation(block: true, alt: "3 c plus 1 minus the fraction 3 over 2 c")[$3 c + 1 − frac(3, 2 c)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "the fraction 10 d squared minus 5 d minus 2 over 5 d .")[$frac(10 d^(2) − 5 d − 2, 5 d) .$]

#solutionbox[
#math.equation(block: true, alt: "2 d minus 1 minus the fraction 2 over 5 d")[$2 d − 1 − frac(2, 5 d)$]
]
]


=== Divide a Polynomial by a Binomial

To divide a polynomial by a #strong[binomial], we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([We write the long division], [#figure(figph[The long division of 875 by 25.], alt: "The long division of 875 by 25.", caption: none)]),
  [We divide the first two digits, 87, by 25.], [#figure(figph[25 fits into 87 three times. 3 is written above the second digit of 875 in the long division bracket.], alt: "25 fits into 87 three times. 3 is written above the second digit of 875 in the long division bracket.", caption: none)],
  [We multiply 3 times 25 and write the product under the 87.], [#figure(figph[The product of 3 and 25 is 75, which is written below the first two digits of 875 in the long division bracket.], alt: "The product of 3 and 25 is 75, which is written below the first two digits of 875 in the long division bracket.", caption: none)],
  [Now we subtract 75 from 87.], [#figure(figph[87 minus 75 is 12, which is written under 75.], alt: "87 minus 75 is 12, which is written under 75.", caption: none)],
  [Then we bring down the third digit of the dividend, 5.], [#figure(figph[The 5 in 875 is brought down next to the 12, making 125.], alt: "The 5 in 875 is brought down next to the 12, making 125.", caption: none)],
  [Repeat the process, dividing 25 into 125.], [#figure(figph[25 fits into 125 five times. 5 is written to the right of the 3 on top of the long division bracket. 5 times 25 is 125. 125 minus 125 is zero. There is zero remainder, so 25 fits into 125 exactly five times. 875 divided by 25 equals 35.], alt: "25 fits into 125 five times. 5 is written to the right of the 3 on top of the long division bracket. 5 times 25 is 125. 125 minus 125 is zero. There is zero remainder, so 25 fits into 125 exactly five times. 875 divided by 25 equals 35.", caption: none)],
))

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

#math.equation(block: true, alt: "35 times 25; 875 ✓")[$35 · 25 \ 875 #h(0.2em) "✓"$]Now we will divide a #strong[trinomial] by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

#examplebox("Example 8")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis x squared plus 9 x plus 20 close parenthesis divided by open parenthesis x plus 5 close parenthesis .")[$( x^(2) + 9 x + 20 ) ÷ ( x + 5 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A trinomial, x squared plus 9 x plus 20, divided by a binomial, x plus 5.], alt: "A trinomial, x squared plus 9 x plus 20, divided by a binomial, x plus 5.", caption: none)]),
  [Write it as a long division problem.], [],
  [Be sure the dividend is in standard form.], [#figure(figph[The long division of x squared plus 9 x plus 20 by x plus 5], alt: "The long division of x squared plus 9 x plus 20 by x plus 5", caption: none)],
  [Divide #emph[x]#super[2] by #emph[x]. It may help to ask yourself, "What do I need to multiply #emph[x] by to get #emph[x]#super[2]?"], [],
  [Put the answer, #emph[x], in the quotient over the #emph[x] term.], [#figure(figph[x fits into x squared x times. x is written above the second term of x squared plus 9 x plus 20 in the long division bracket.], alt: "x fits into x squared x times. x is written above the second term of x squared plus 9 x plus 20 in the long division bracket.", caption: none)],
  [Multiply #emph[x] times #emph[x] + 5. Line up the like terms under the dividend.], [#figure(figph[The product of x and x plus 5 is x squared plus 5 x, which is written below the first two terms of x squared plus 9x plus 20 in the long division bracket.], alt: "The product of x and x plus 5 is x squared plus 5 x, which is written below the first two terms of x squared plus 9x plus 20 in the long division bracket.", caption: none)],
  [Subtract #emph[x]#super[2] + 5#emph[x] from #emph[x]#super[2] + 9#emph[x].], [],
  [#figure(figph[You may find it easier to change the signs and then add.], alt: "You may find it easier to change the signs and then add.", caption: none)

 #linebreak() Then bring down the last term, 20.], [#figure(figph[The sum of x squared plus 9 x and negative x squared plus negative 5 x is 4 x, which is written underneath the negative 5 x. The third term in x squared plus 9 x plus 20 is brought down next to 4 x, making 4 x plus 20.], alt: "The sum of x squared plus 9 x and negative x squared plus negative 5 x is 4 x, which is written underneath the negative 5 x. The third term in x squared plus 9 x plus 20 is brought down next to 4 x, making 4 x plus 20.", caption: none)],
  [Divide 4#emph[x] by #emph[x]. It may help to ask yourself, "What do I need to #linebreak() multiply #emph[x] by to get 4#emph[x]?"], [],
  [Put the answer, 4, in the quotient over the constant term.], [#figure(figph[4 x divided by x is 4. Plus 4 is written on top of the long division bracket, next to x and above the 20 in x squared plus 9 x plus 20.], alt: "4 x divided by x is 4. Plus 4 is written on top of the long division bracket, next to x and above the 20 in x squared plus 9 x plus 20.", caption: none)],
  [Multiply 4 times #emph[x] + 5.], [#figure(figph[x plus 5 times 4 is 4 x plus 20, which is written under the first 4 x plus 20.], alt: "x plus 5 times 4 is 4 x plus 20, which is written under the first 4 x plus 20.", caption: none)],
  [Subtract 4#emph[x] + 20 from 4#emph[x] + 20.], [#figure(figph[4 x plus 20 minus 4 x plus 20 is 0. The remainder is 0. x squared plus 9 x plus 20 divided by x plus 5 equals x plus 4.], alt: "4 x plus 20 minus 4 x plus 20 is 0. The remainder is 0. x squared plus 9 x plus 20 divided by x plus 5 equals x plus 4.", caption: none)],
  [Check:], [],
  [Multiply the quotient by the divisor.], [],
  [(#emph[x] + 4)(#emph[x] + 5)], [],
  [You should get the dividend.], [],
  [#emph[x]#super[2] + 9#emph[x] + 20✓], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis y squared plus 10 y plus 21 close parenthesis divided by open parenthesis y plus 3 close parenthesis .")[$( y^(2) + 10 y + 21 ) ÷ ( y + 3 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y plus 7")[$y + 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis m squared plus 9 m plus 20 close parenthesis divided by open parenthesis m plus 4 close parenthesis .")[$( m^(2) + 9 m + 20 ) ÷ ( m + 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "m plus 5")[$m + 5$]
]
]

When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.

#examplebox("Example 9")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis 2 x squared minus 5 x minus 3 close parenthesis divided by open parenthesis x minus 3 close parenthesis .")[$( 2 x^(2) − 5 x − 3 ) ÷ ( x − 3 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A trinomial, 2 x squared minus 5 x minus 3, divided by a binomial, x minus 3.], alt: "A trinomial, 2 x squared minus 5 x minus 3, divided by a binomial, x minus 3.", caption: none)]),
  [Write it as a long division problem.], [],
  [Be sure the dividend is in standard form.], [#figure(figph[The long division of 2 x squared minus 5 x minus 3 by x minus 3.], alt: "The long division of 2 x squared minus 5 x minus 3 by x minus 3.", caption: none)],
  [Divide 2#emph[x]#super[2] by #emph[x]. #linebreak() Put the answer, 2#emph[x], in the quotient over the #emph[x] term.], [#figure(figph[x fits into 2 x squared 2 x times. 2 x is written above the second term of 2 x squared minus 5 x minus 3 in the long division bracket.], alt: "x fits into 2 x squared 2 x times. 2 x is written above the second term of 2 x squared minus 5 x minus 3 in the long division bracket.", caption: none)],
  [Multiply 2#emph[x] times #emph[x] − 3. Line up the like terms under the dividend.], [#figure(figph[The product of 2 x and x minus 3 is 2 x squared minus 6 x, which is written below the first two terms of 2 x squared minus 5 x minus 3 in the long division bracket.], alt: "The product of 2 x and x minus 3 is 2 x squared minus 6 x, which is written below the first two terms of 2 x squared minus 5 x minus 3 in the long division bracket.", caption: none)],
  [Subtract 2#emph[x]#super[2] − 6#emph[x] from 2#emph[x]#super[2] − 5#emph[x]. #linebreak() Change the signs and then add. #linebreak() Then bring down the last term.], [#figure(figph[The sum of 2 x squared minus 5 x and negative 2 x squared plus 6 x is x, which is written underneath the 6 x. The third term in 2 x squared minus 5 x minus 3 is brought down next to x, making x minus 3.], alt: "The sum of 2 x squared minus 5 x and negative 2 x squared plus 6 x is x, which is written underneath the 6 x. The third term in 2 x squared minus 5 x minus 3 is brought down next to x, making x minus 3.", caption: none)],
  [Divide  #emph[x] by #emph[x]. #linebreak() Put the answer, 1, in the quotient over the constant term.], [#figure(figph[Plus 1 is written on top of the long division bracket, next to 2 x and above the minus 3 in 2 x squared minus 5 x minus 3.], alt: "Plus 1 is written on top of the long division bracket, next to 2 x and above the minus 3 in 2 x squared minus 5 x minus 3.", caption: none)],
  [Multiply 1 times #emph[x] − 3.], [#figure(figph[x minus 3 times 1 is x minus 3, which is written under the first x minus 3.], alt: "x minus 3 times 1 is x minus 3, which is written under the first x minus 3.", caption: none)],
  [Subtract #emph[x] − 3 from #emph[x] − 3 by changing the signs and adding.], [#figure(figph[The binomial x minus 3 minus the binomial negative x plus 3 is 0. The remainder is 0. 2 x squared minus 5 x minus 3 divided by x minus 3 equals 2 x plus 1.], alt: "The binomial x minus 3 minus the binomial negative x plus 3 is 0. The remainder is 0. 2 x squared minus 5 x minus 3 divided by x minus 3 equals 2 x plus 1.", caption: none)],
  [To check, multiply (#emph[x] − 3)(2#emph[x] + 1).], [],
  [The result should be 2#emph[x]#super[2] − 5#emph[x] − 3.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 2 x squared minus 3 x minus 20 close parenthesis divided by open parenthesis x minus 4 close parenthesis .")[$( 2 x^(2) − 3 x − 20 ) ÷ ( x − 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "2 x plus 5")[$2 x + 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 3 x squared minus 16 x minus 12 close parenthesis divided by open parenthesis x minus 6 close parenthesis .")[$( 3 x^(2) − 16 x − 12 ) ÷ ( x − 6 ) .$]

#solutionbox[
#math.equation(block: true, alt: "3 x plus 2")[$3 x + 2$]
]
]

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In , we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

#examplebox("Example 10")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis x cubed minus x squared plus x plus 4 close parenthesis divided by open parenthesis x plus 1 close parenthesis .")[$( x^(3) − x^(2) + x + 4 ) ÷ ( x + 1 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A polynomial, x cubed minus x squared plus x plus 4, divided by another polynomial, x plus 1.], alt: "A polynomial, x cubed minus x squared plus x plus 4, divided by another polynomial, x plus 1.", caption: none)]),
  [Write it as a long division problem.], [],
  [Be sure the dividend is in standard form.], [#figure(figph[The long division of x cubed minus x squared plus x plus 4 by x plus 1.], alt: "The long division of x cubed minus x squared plus x plus 4 by x plus 1.", caption: none)],
  [Divide #emph[x]#super[3] by #emph[x]. #linebreak() Put the answer, #emph[x]#super[2], in the quotient over the #emph[x]#super[2] term. #linebreak() Multiply #emph[x]#super[2] times #emph[x] + 1. Line up the like terms under the dividend.], [#figure(figph[x fits into x squared x times. x is written above the second term of x cubed minus x squared plus x plus 4 in the long division bracket.], alt: "x fits into x squared x times. x is written above the second term of x cubed minus x squared plus x plus 4 in the long division bracket.", caption: none)],
  [Subtract #emph[x]#super[3] + #emph[x]#super[2] from #emph[x]#super[3] − #emph[x]#super[2] by changing the signs and adding. #linebreak() Then bring down the next term.], [#figure(figph[The sum of x cubed minus x squared and negative x cubed plus negative x squared is negative 2 x squared, which is written underneath the negative x squared. The next term in x cubed minus x squared plus x plus 4 is brought down next to negative 2 x squared, making negative 2 x squared plus x.], alt: "The sum of x cubed minus x squared and negative x cubed plus negative x squared is negative 2 x squared, which is written underneath the negative x squared. The next term in x cubed minus x squared plus x plus 4 is brought down next to negative 2 x squared, making negative 2 x squared plus x.", caption: none)],
  [Divide −2#emph[x]#super[2] by #emph[x]. #linebreak() Put the answer, −2#emph[x], in the quotient over the #emph[x] term. #linebreak() Multiply −2#emph[x] times #emph[x] + 1. Line up the like terms under the dividend.], [#figure(figph[Minus 2 x is written on top of the long division bracket, next to x squared and above the x in x cubed minus x squared plus x plus 4. Negative 2 x squared minus 2 x is written under negative 2 x squared plus x.], alt: "Minus 2 x is written on top of the long division bracket, next to x squared and above the x in x cubed minus x squared plus x plus 4. Negative 2 x squared minus 2 x is written under negative 2 x squared plus x.", caption: none)],
  [Subtract −2#emph[x]#super[2] − 2#emph[x] from −2#emph[x]#super[2] + #emph[x] by changing the signs and adding. #linebreak() Then bring down the last term.], [#figure(figph[The sum of negative 2 x squared plus x and 2 x squared plus 2 x is found to be 3 x. The last term in x cubed minus x squared plus x plus 4 is brought down, making 3 x plus 4.], alt: "The sum of negative 2 x squared plus x and 2 x squared plus 2 x is found to be 3 x. The last term in x cubed minus x squared plus x plus 4 is brought down, making 3 x plus 4.", caption: none)],
  [Divide 3#emph[x] by #emph[x]. #linebreak() Put the answer, 3, in the quotient over the constant term. #linebreak() Multiply 3 times #emph[x] + 1. Line up the like terms under the dividend.], [#figure(figph[Plus 3 is written on top of the long division bracket, above the 4 in x cubed minus x squared plus x plus 4. 3 x plus 3 is written under 3 x plus 4.], alt: "Plus 3 is written on top of the long division bracket, above the 4 in x cubed minus x squared plus x plus 4. 3 x plus 3 is written under 3 x plus 4.", caption: none)],
  [Subtract 3#emph[x] + 3 from 3#emph[x] + 4 by changing the signs and adding. #linebreak() Write the remainder as a fraction with the divisor as the denominator.], [#figure(figph[The sum of 3 x plus 4 and negative 3 x plus negative 3 is 1. Therefore, the polynomial x cubed minus x squared plus x plus 4, divided by the binomial x plus 1, equals x squared minus 2 x plus the fraction 1 over x plus 1.], alt: "The sum of 3 x plus 4 and negative 3 x plus negative 3 is 1. Therefore, the polynomial x cubed minus x squared plus x plus 4, divided by the binomial x plus 1, equals x squared minus 2 x plus the fraction 1 over x plus 1.", caption: none)],
  [To check, multiply #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis open parenthesis x squared minus 2 x plus 3 plus the fraction 1 over x plus 1 close parenthesis .")[$( x + 1 ) ( x^(2) − 2 x + 3 + frac(1, x + 1) ) .$] #linebreak() The result should be #math.equation(block: false, alt: "x cubed minus x squared plus x plus 4")[$x^(3) − x^(2) + x + 4$].], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x cubed plus 5 x squared plus 8 x plus 6 close parenthesis divided by open parenthesis x plus 2 close parenthesis .")[$( x^(3) + 5 x^(2) + 8 x + 6 ) ÷ ( x + 2 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus 3 x plus 2 plus the fraction 2 over x plus 2")[$x^(2) + 3 x + 2 + frac(2, x + 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 2 x cubed plus 8 x squared plus x minus 8 close parenthesis divided by open parenthesis x plus 1 close parenthesis .")[$( 2 x^(3) + 8 x^(2) + x − 8 ) ÷ ( x + 1 ) .$]

#solutionbox[
#math.equation(block: true, alt: "2 x squared plus 6 x minus 5 minus the fraction 3 over x plus 1")[$2 x^(2) + 6 x − 5 − frac(3, x + 1)$]
]
]

Look back at the dividends in , , and . The terms were written in descending order of degrees, and there were no missing degrees. The dividend in  will be #math.equation(block: false, alt: "x to the power 4 minus x squared plus 5 x minus 2")[$x^(4) − x^(2) + 5 x − 2$]. It is missing an #math.equation(block: false, alt: "x cubed")[$x^(3)$] term. We will add in #math.equation(block: false, alt: "0 x cubed")[$0 x^(3)$] as a placeholder.

#examplebox("Example 11")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis x to the power 4 minus x squared plus 5 x minus 2 close parenthesis divided by open parenthesis x plus 2 close parenthesis .")[$( x^(4) − x^(2) + 5 x − 2 ) ÷ ( x + 2 ) .$]

#solutionbox[
Notice that there is no #math.equation(block: false, alt: "x cubed")[$x^(3)$] term in the dividend. We will add #math.equation(block: false, alt: "0 x cubed")[$0 x^(3)$] as a placeholder.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A polynomial, x to the fourth power minus x squared minus 5 x minus 2, divided by another polynomial, x plus 2.], alt: "A polynomial, x to the fourth power minus x squared minus 5 x minus 2, divided by another polynomial, x plus 2.", caption: none)]),
  [Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.], [#figure(figph[The long division of x to the fourth power plus 0 x cubed minus x squared minus 5 x minus 2 by x plus 2.], alt: "The long division of x to the fourth power plus 0 x cubed minus x squared minus 5 x minus 2 by x plus 2.", caption: none)],
  [Divide #emph[x]#super[4] by #emph[x]. #linebreak() Put the answer, #emph[x]#super[3], in the quotient over the #emph[x]#super[3] term. #linebreak() Multiply #emph[x]#super[3] times #emph[x] + 2. Line up the like terms. #linebreak() Subtract and then bring down the next term.], [#figure(figph[x cubed is written on top of the long division bracket above the x cubed term in the dividend. Below the first two terms of the dividend x to the fourth power plus 2 x cubed is subtracted to give negative 2 x cubed minus x squared. A note next to the division reads “It may be helpful to change the signs and add.”], alt: "x cubed is written on top of the long division bracket above the x cubed term in the dividend. Below the first two terms of the dividend x to the fourth power plus 2 x cubed is subtracted to give negative 2 x cubed minus x squared. A note next to the division reads “It may be helpful to change the signs and add.”", caption: none)],
  [Divide −2#emph[x]#super[3] by #emph[x]. #linebreak() Put the answer, −2#emph[x]#super[2], in the quotient over the #emph[x]#super[2] term. #linebreak() Multiply −2#emph[x]#super[2] times #emph[x] + 1. Line up the like terms. #linebreak() Subtract and bring down the next term.], [#figure(figph[x cubed minus 2 x squared is written on top of the long division bracket. At the bottom of the long division negative 2 x cubed minus 4 x squared is subtracted to give 3 x squared plus 5 x. A note reads “It may be helpful to change the signs and add.”], alt: "x cubed minus 2 x squared is written on top of the long division bracket. At the bottom of the long division negative 2 x cubed minus 4 x squared is subtracted to give 3 x squared plus 5 x. A note reads “It may be helpful to change the signs and add.”", caption: none)],
  [Divide 3#emph[x]#super[2] by #emph[x]. #linebreak() Put the answer, 3#emph[x], in the quotient over the #emph[x] term. #linebreak() Multiply 3#emph[x] times #emph[x] + 1. Line up the like terms. #linebreak() Subtract and bring down the next term.], [#figure(figph[x cubed minus 2 x squared plus 3 x is written on top of the long division bracket. At the bottom of the long division 3 x squared plus 6 x is subtracted to give negative x minus 2. A note reads “It may be helpful to change the signs and add.”], alt: "x cubed minus 2 x squared plus 3 x is written on top of the long division bracket. At the bottom of the long division 3 x squared plus 6 x is subtracted to give negative x minus 2. A note reads “It may be helpful to change the signs and add.”", caption: none)],
  [Divide −#emph[x] by #emph[x]. #linebreak() Put the answer, −1, in the quotient over the constant term. #linebreak() Multiply −1 times #emph[x] + 1. Line up the like terms. #linebreak() Change the signs, add.], [#figure(figph[x cubed minus 2 x squared plus 3 x minus 1 is written on top of the long division bracket. At the bottom of the long division negative x minus 2 is subtract to give 0. A note reads “It may be helpful to change the signs and add.” The polynomial x to the fourth power minus x squared plus 5 x minus 2, divided by the binomial x plus 2 equals the polynomial x cubed minus 2 x squared plus 3 x minus 1.], alt: "x cubed minus 2 x squared plus 3 x minus 1 is written on top of the long division bracket. At the bottom of the long division negative x minus 2 is subtract to give 0. A note reads “It may be helpful to change the signs and add.” The polynomial x to the fourth power minus x squared plus 5 x minus 2, divided by the binomial x plus 2 equals the polynomial x cubed minus 2 x squared plus 3 x minus 1.", caption: none)],
  [To check, multiply #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x cubed minus 2 x squared plus 3 x minus 1 close parenthesis")[$( x + 2 ) ( x^(3) − 2 x^(2) + 3 x − 1 )$].], [],
  [The result should be #math.equation(block: false, alt: "x to the power 4 minus x squared plus 5 x minus 2")[$x^(4) − x^(2) + 5 x − 2$].], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x cubed plus 3 x plus 14 close parenthesis divided by open parenthesis x plus 2 close parenthesis .")[$( x^(3) + 3 x + 14 ) ÷ ( x + 2 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x squared minus 2 x plus 7")[$x^(2) − 2 x + 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x to the power 4 minus 3 x cubed minus 1000 close parenthesis divided by open parenthesis x plus 5 close parenthesis .")[$( x^(4) − 3 x^(3) − 1000 ) ÷ ( x + 5 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x cubed minus 8 x squared plus 40 x minus 200")[$x^(3) − 8 x^(2) + 40 x − 200$]
]
]

In , we will divide by #math.equation(block: false, alt: "2 a minus 3")[$2 a − 3$]. As we divide we will have to consider the constants as well as the variables.

#examplebox("Example 12")[][
Find the quotient: #math.equation(block: false, alt: "open parenthesis 8 a cubed plus 27 close parenthesis divided by open parenthesis 2 a plus 3 close parenthesis .")[$( 8 a^(3) + 27 ) ÷ ( 2 a + 3 ) .$]

#solutionbox[
This time we will show the division all in one step. We need to add two placeholders in order to divide.

#figure(figph[The figure shows the long division of 8 a cubed plus 27 by 2 a plus 3. In the long division bracket, placeholders 0 a squared and 0 a are added into the polynomial. On the first line under the dividend 8 a cubed plus 12 a squared is subtracted. To the right, an arrow indicates that this value came from multiplying 4 a squared by 2 a plus 3. The subtraction gives negative 12 a squared plus 0 a. From this negative 12 a squared minus 18 a is subtracted. To the right, an arrow indicates that this value came from multiplying 6 a by 2 a plus 3. The subtraction give 18 a plus 27. From this 18 a plus 27 is subtracted. To the right, an arrow indicates that this value came from multiplying 9 by 2 a plus 3. The result is 0.], alt: "The figure shows the long division of 8 a cubed plus 27 by 2 a plus 3. In the long division bracket, placeholders 0 a squared and 0 a are added into the polynomial. On the first line under the dividend 8 a cubed plus 12 a squared is subtracted. To the right, an arrow indicates that this value came from multiplying 4 a squared by 2 a plus 3. The subtraction gives negative 12 a squared plus 0 a. From this negative 12 a squared minus 18 a is subtracted. To the right, an arrow indicates that this value came from multiplying 6 a by 2 a plus 3. The subtraction give 18 a plus 27. From this 18 a plus 27 is subtracted. To the right, an arrow indicates that this value came from multiplying 9 by 2 a plus 3. The result is 0.", caption: none)

To check, multiply #math.equation(block: false, alt: "open parenthesis 2 a plus 3 close parenthesis open parenthesis 4 a squared minus 6 a plus 9 close parenthesis")[$( 2 a + 3 ) ( 4 a^(2) − 6 a + 9 )$].

The result should be #math.equation(block: false, alt: "8 a cubed plus 27")[$8 a^(3) + 27$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis x cubed minus 64 close parenthesis divided by open parenthesis x minus 4 close parenthesis .")[$( x^(3) − 64 ) ÷ ( x − 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus 4 x plus 16")[$x^(2) + 4 x + 16$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "open parenthesis 125 x cubed minus 8 close parenthesis divided by open parenthesis 5 x minus 2 close parenthesis .")[$( 125 x^(3) − 8 ) ÷ ( 5 x − 2 ) .$]

#solutionbox[
#math.equation(block: true, alt: "25 x squared plus 10 x plus 4")[$25 x^(2) + 10 x + 4$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with dividing polynomials:

- #link("https://openstax.org/l/25DividePolyMo1")[Divide a Polynomial by a Monomial]
- #link("https://openstax.org/l/25DividePolyMo2")[Divide a Polynomial by a Monomial 2]
- #link("https://openstax.org/l/25DividePolyBin")[Divide Polynomial by Binomial]
]


=== Key Concepts

- #strong[Fraction Addition]
- If #math.equation(block: false, alt: "a , b , and c")[$a , b , #h(0.2em) "and" #h(0.2em) c$] are numbers where #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], then #linebreak() #math.equation(block: false, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c and the fraction a plus b over c equals the fraction a over c plus the fraction b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c) #h(0.5em) "and" #h(0.5em) frac(a + b, c) = frac(a, c) + frac(b, c)$] #linebreak()  #linebreak()
- #strong[Division of a Polynomial by a Monomial]
- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.


==== Practice Makes Perfect

In the following exercises, divide each polynomial by the monomial.

#math.equation(block: true, alt: "the fraction 45 y plus 36 over 9")[$frac(45 y + 36, 9)$]

#math.equation(block: true, alt: "the fraction 30 b plus 75 over 5")[$frac(30 b + 75, 5)$]

#solutionbox[
#math.equation(block: true, alt: "6 b plus 15")[$6 b + 15$]
]

#math.equation(block: true, alt: "the fraction 8 d squared minus 4 d over 2")[$frac(8 d^(2) − 4 d, 2)$]

#math.equation(block: true, alt: "the fraction 42 x squared minus 14 x over 7")[$frac(42 x^(2) − 14 x, 7)$]

#solutionbox[
#math.equation(block: true, alt: "6 x squared minus 2 x")[$6 x^(2) − 2 x$]
]

#math.equation(block: true, alt: "open parenthesis 16 y squared minus 20 y close parenthesis divided by 4 y")[$( 16 y^(2) − 20 y ) ÷ 4 y$]

#math.equation(block: true, alt: "open parenthesis 55 w squared minus 10 w close parenthesis divided by 5 w")[$( 55 w^(2) − 10 w ) ÷ 5 w$]

#solutionbox[
#math.equation(block: true, alt: "11 w minus 2")[$11 w − 2$]
]

#math.equation(block: true, alt: "open parenthesis 9 n to the power 4 plus 6 n cubed close parenthesis divided by 3 n")[$( 9 n^(4) + 6 n^(3) ) ÷ 3 n$]

#math.equation(block: true, alt: "open parenthesis 8 x cubed plus 6 x squared close parenthesis divided by 2 x")[$( 8 x^(3) + 6 x^(2) ) ÷ 2 x$]

#solutionbox[
#math.equation(block: true, alt: "4 x squared plus 3 x")[$4 x^(2) + 3 x$]
]

#math.equation(block: true, alt: "the fraction 18 y squared minus 12 y over −6")[$frac(18 y^(2) − 12 y, −6)$]

#math.equation(block: true, alt: "the fraction 20 b squared minus 12 b over −4")[$frac(20 b^(2) − 12 b, −4)$]

#solutionbox[
#math.equation(block: true, alt: "−5 b squared plus 3 b")[$−5 b^(2) + 3 b$]
]

#math.equation(block: true, alt: "the fraction 35 a to the power 4 plus 65 a squared over −5")[$frac(35 a^(4) + 65 a^(2), −5)$]

#math.equation(block: true, alt: "the fraction 51 m to the power 4 plus 72 m cubed over −3")[$frac(51 m^(4) + 72 m^(3), −3)$]

#solutionbox[
#math.equation(block: true, alt: "−17 m to the power 4 minus 24 m cubed")[$−17 m^(4) − 24 m^(3)$]
]

#math.equation(block: true, alt: "the fraction 310 y to the power 4 minus 200 y cubed over 5 y squared")[$frac(310 y^(4) − 200 y^(3), 5 y^(2))$]

#math.equation(block: true, alt: "the fraction 412 z to the power 8 minus 48 z to the power 5 over 4 z cubed")[$frac(412 z^(8) − 48 z^(5), 4 z^(3))$]

#solutionbox[
#math.equation(block: true, alt: "103 z to the power 5 minus 12 z squared")[$103 z^(5) − 12 z^(2)$]
]

#math.equation(block: true, alt: "the fraction 46 x cubed plus 38 x squared over 2 x squared")[$frac(46 x^(3) + 38 x^(2), 2 x^(2))$]

#math.equation(block: true, alt: "the fraction 51 y to the power 4 plus 42 y squared over 3 y squared")[$frac(51 y^(4) + 42 y^(2), 3 y^(2))$]

#solutionbox[
#math.equation(block: true, alt: "17 y squared plus 14")[$17 y^(2) + 14$]
]

#math.equation(block: true, alt: "open parenthesis 24 p squared minus 33 p close parenthesis divided by open parenthesis −3 p close parenthesis")[$( 24 p^(2) − 33 p ) ÷ ( −3 p )$]

#math.equation(block: true, alt: "open parenthesis 35 x to the power 4 minus 21 x close parenthesis divided by open parenthesis −7 x close parenthesis")[$( 35 x^(4) − 21 x ) ÷ ( −7 x )$]

#solutionbox[
#math.equation(block: true, alt: "−5 x cubed plus 3")[$−5 x^(3) + 3$]
]

#math.equation(block: true, alt: "open parenthesis 63 m to the power 4 minus 42 m cubed close parenthesis divided by open parenthesis −7 m squared close parenthesis")[$( 63 m^(4) − 42 m^(3) ) ÷ ( −7 m^(2) )$]

#math.equation(block: true, alt: "open parenthesis 48 y to the power 4 minus 24 y cubed close parenthesis divided by open parenthesis −8 y squared close parenthesis")[$( 48 y^(4) − 24 y^(3) ) ÷ ( −8 y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "−6 y squared plus 3 y")[$−6 y^(2) + 3 y$]
]

#math.equation(block: true, alt: "open parenthesis 63 a squared b cubed plus 72 a b to the power 4 close parenthesis divided by open parenthesis 9 a b close parenthesis")[$( 63 a^(2) b^(3) + 72 a b^(4) ) ÷ ( 9 a b )$]

#math.equation(block: true, alt: "open parenthesis 45 x cubed y to the power 4 plus 60 x y squared close parenthesis divided by open parenthesis 5 x y close parenthesis")[$( 45 x^(3) y^(4) + 60 x y^(2) ) ÷ ( 5 x y )$]

#solutionbox[
#math.equation(block: true, alt: "9 x squared y cubed plus 12 y")[$9 x^(2) y^(3) + 12 y$]
]

#math.equation(block: true, alt: "the fraction 52 p to the power 5 q to the power 4 plus 36 p to the power 4 q cubed minus 64 p cubed q squared over 4 p squared q")[$frac(52 p^(5) q^(4) + 36 p^(4) q^(3) − 64 p^(3) q^(2), 4 p^(2) q)$]

#math.equation(block: true, alt: "the fraction 49 c squared d squared minus 70 c cubed d cubed minus 35 c squared d to the power 4 over 7 c d squared")[$frac(49 c^(2) d^(2) − 70 c^(3) d^(3) − 35 c^(2) d^(4), 7 c d^(2))$]

#solutionbox[
#math.equation(block: true, alt: "7 c minus 10 c squared d minus 5 c d squared")[$7 c − 10 c^(2) d − 5 c d^(2)$]
]

#math.equation(block: true, alt: "the fraction 66 x cubed y squared minus 110 x squared y cubed minus 44 x to the power 4 y cubed over 11 x squared y squared")[$frac(66 x^(3) y^(2) − 110 x^(2) y^(3) − 44 x^(4) y^(3), 11 x^(2) y^(2))$]

#math.equation(block: true, alt: "the fraction 72 r to the power 5 s squared plus 132 r to the power 4 s cubed minus 96 r cubed s to the power 5 over 12 r squared s squared")[$frac(72 r^(5) s^(2) + 132 r^(4) s^(3) − 96 r^(3) s^(5), 12 r^(2) s^(2))$]

#solutionbox[
#math.equation(block: true, alt: "6 r cubed plus 11 r squared s minus 8 r s cubed")[$6 r^(3) + 11 r^(2) s − 8 r s^(3)$]
]

#math.equation(block: true, alt: "the fraction 4 w squared plus 2 w minus 5 over 2 w")[$frac(4 w^(2) + 2 w − 5, 2 w)$]

#math.equation(block: true, alt: "the fraction 12 q squared plus 3 q minus 1 over 3 q")[$frac(12 q^(2) + 3 q − 1, 3 q)$]

#solutionbox[
#math.equation(block: true, alt: "4 q plus 1 minus the fraction 1 over 3 q")[$4 q + 1 − frac(1, 3 q)$]
]

#math.equation(block: true, alt: "the fraction 10 x squared plus 5 x minus 4 over −5 x")[$frac(10 x^(2) + 5 x − 4, −5 x)$]

#math.equation(block: true, alt: "the fraction 20 y squared plus 12 y minus 1 over −4 y")[$frac(20 y^(2) + 12 y − 1, −4 y)$]

#solutionbox[
#math.equation(block: true, alt: "−5 y minus 3 plus the fraction 1 over 4 y")[$−5 y − 3 + frac(1, 4 y)$]
]

#math.equation(block: true, alt: "the fraction 36 p cubed plus 18 p squared minus 12 p over 6 p squared")[$frac(36 p^(3) + 18 p^(2) − 12 p, 6 p^(2))$]

#math.equation(block: true, alt: "the fraction 63 a cubed minus 108 a squared plus 99 a over 9 a squared")[$frac(63 a^(3) − 108 a^(2) + 99 a, 9 a^(2))$]

#solutionbox[
#math.equation(block: true, alt: "7 a minus 12 plus the fraction 11 over a")[$7 a − 12 + frac(11, a)$]
]

#strong[Divide a Polynomial by a Binomial]

In the following exercises, divide each polynomial by the binomial.

#math.equation(block: true, alt: "open parenthesis y squared plus 7 y plus 12 close parenthesis divided by open parenthesis y plus 3 close parenthesis")[$( y^(2) + 7 y + 12 ) ÷ ( y + 3 )$]

#math.equation(block: true, alt: "open parenthesis d squared plus 8 d plus 12 close parenthesis divided by open parenthesis d plus 2 close parenthesis")[$( d^(2) + 8 d + 12 ) ÷ ( d + 2 )$]

#solutionbox[
#math.equation(block: true, alt: "d plus 6")[$d + 6$]
]

#math.equation(block: true, alt: "open parenthesis x squared minus 3 x minus 10 close parenthesis divided by open parenthesis x plus 2 close parenthesis")[$( x^(2) − 3 x − 10 ) ÷ ( x + 2 )$]

#math.equation(block: true, alt: "open parenthesis a squared minus 2 a minus 35 close parenthesis divided by open parenthesis a plus 5 close parenthesis")[$( a^(2) − 2 a − 35 ) ÷ ( a + 5 )$]

#solutionbox[
#math.equation(block: true, alt: "a minus 7")[$a − 7$]
]

#math.equation(block: true, alt: "open parenthesis t squared minus 12 t plus 36 close parenthesis divided by open parenthesis t minus 6 close parenthesis")[$( t^(2) − 12 t + 36 ) ÷ ( t − 6 )$]

#math.equation(block: true, alt: "open parenthesis x squared minus 14 x plus 49 close parenthesis divided by open parenthesis x minus 7 close parenthesis")[$( x^(2) − 14 x + 49 ) ÷ ( x − 7 )$]

#solutionbox[
#math.equation(block: true, alt: "x minus 7")[$x − 7$]
]

#math.equation(block: true, alt: "open parenthesis 6 m squared minus 19 m minus 20 close parenthesis divided by open parenthesis m minus 4 close parenthesis")[$( 6 m^(2) − 19 m − 20 ) ÷ ( m − 4 )$]

#math.equation(block: true, alt: "open parenthesis 4 x squared minus 17 x minus 15 close parenthesis divided by open parenthesis x minus 5 close parenthesis")[$( 4 x^(2) − 17 x − 15 ) ÷ ( x − 5 )$]

#solutionbox[
#math.equation(block: true, alt: "4 x plus 3")[$4 x + 3$]
]

#math.equation(block: true, alt: "open parenthesis q squared plus 2 q plus 20 close parenthesis divided by open parenthesis q plus 6 close parenthesis")[$( q^(2) + 2 q + 20 ) ÷ ( q + 6 )$]

#math.equation(block: true, alt: "open parenthesis p squared plus 11 p plus 16 close parenthesis divided by open parenthesis p plus 8 close parenthesis")[$( p^(2) + 11 p + 16 ) ÷ ( p + 8 )$]

#solutionbox[
#math.equation(block: true, alt: "p plus 3 minus the fraction 8 over p plus 8")[$p + 3 − frac(8, p + 8)$]
]

#math.equation(block: true, alt: "open parenthesis y squared minus 3 y minus 15 close parenthesis divided by open parenthesis y minus 8 close parenthesis")[$( y^(2) − 3 y − 15 ) ÷ ( y − 8 )$]

#math.equation(block: true, alt: "open parenthesis x squared plus 2 x minus 30 close parenthesis divided by open parenthesis x minus 5 close parenthesis")[$( x^(2) + 2 x − 30 ) ÷ ( x − 5 )$]

#solutionbox[
#math.equation(block: true, alt: "x plus 7 plus the fraction 5 over x minus 5")[$x + 7 + frac(5, x − 5)$]
]

#math.equation(block: true, alt: "open parenthesis 3 b cubed plus b squared plus 2 close parenthesis divided by open parenthesis b plus 1 close parenthesis")[$( 3 b^(3) + b^(2) + 2 ) ÷ ( b + 1 )$]

#math.equation(block: true, alt: "open parenthesis 2 n cubed minus 10 n plus 24 close parenthesis divided by open parenthesis n plus 3 close parenthesis")[$( 2 n^(3) − 10 n + 24 ) ÷ ( n + 3 )$]

#solutionbox[
#math.equation(block: true, alt: "2 n squared minus 6 n plus 8")[$2 n^(2) − 6 n + 8$]
]

#math.equation(block: true, alt: "open parenthesis 2 y cubed minus 6 y minus 36 close parenthesis divided by open parenthesis y minus 3 close parenthesis")[$( 2 y^(3) − 6 y − 36 ) ÷ ( y − 3 )$]

#math.equation(block: true, alt: "open parenthesis 7 q cubed minus 5 q minus 2 close parenthesis divided by open parenthesis q minus 1 close parenthesis")[$( 7 q^(3) − 5 q − 2 ) ÷ ( q − 1 )$]

#solutionbox[
#math.equation(block: true, alt: "7 q squared plus 7 q plus 2")[$7 q^(2) + 7 q + 2$]
]

#math.equation(block: true, alt: "open parenthesis z cubed plus 1 close parenthesis divided by open parenthesis z plus 1 close parenthesis")[$( z^(3) + 1 ) ÷ ( z + 1 )$]

#math.equation(block: true, alt: "open parenthesis m cubed plus 1000 close parenthesis divided by open parenthesis m plus 10 close parenthesis")[$( m^(3) + 1000 ) ÷ ( m + 10 )$]

#solutionbox[
#math.equation(block: true, alt: "m squared minus 10 m plus 100")[$m^(2) − 10 m + 100$]
]

#math.equation(block: true, alt: "open parenthesis a cubed minus 125 close parenthesis divided by open parenthesis a minus 5 close parenthesis")[$( a^(3) − 125 ) ÷ ( a − 5 )$]

#math.equation(block: true, alt: "open parenthesis x cubed minus 216 close parenthesis divided by open parenthesis x minus 6 close parenthesis")[$( x^(3) − 216 ) ÷ ( x − 6 )$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus 6 x plus 36")[$x^(2) + 6 x + 36$]
]

#math.equation(block: true, alt: "open parenthesis 64 x cubed minus 27 close parenthesis divided by open parenthesis 4 x minus 3 close parenthesis")[$( 64 x^(3) − 27 ) ÷ ( 4 x − 3 )$]

#math.equation(block: true, alt: "open parenthesis 125 y cubed minus 64 close parenthesis divided by open parenthesis 5 y minus 4 close parenthesis")[$( 125 y^(3) − 64 ) ÷ ( 5 y − 4 )$]

#solutionbox[
#math.equation(block: true, alt: "25 y squared plus 20 x plus 16")[$25 y^(2) + 20 x + 16$]
]


==== Everyday Math

#strong[Average cost] Pictures Plus produces digital albums. The company’s average cost (in dollars) to make #math.equation(block: false, alt: "x")[$x$] albums is given by the expression #math.equation(block: false, alt: "the fraction 7 x plus 500 over x")[$frac(7 x + 500, x)$].

+ ⓐ Find the quotient by dividing the numerator by the denominator.
+ ⓑ What will the average cost (in dollars) be to produce 20 albums?

#strong[Handshakes] At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression #math.equation(block: false, alt: "the fraction n squared minus n over 2")[$frac(n^(2) − n, 2)$], where #math.equation(block: false, alt: "n")[$n$] represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

#solutionbox[
45
]


==== Writing Exercises

James divides #math.equation(block: false, alt: "48 y plus 6")[$48 y + 6$] by 6 this way: #math.equation(block: false, alt: "the fraction 48 y plus 6 over 6 equals 48 y")[$frac(48 y + cancel(6), cancel(6)) = 48 y$]. What is wrong with his reasoning?

Divide #math.equation(block: false, alt: "the fraction 10 x squared plus x minus 12 over 2 x")[$frac(10 x^(2) + x − 12, 2 x)$] and explain with words how you get each term of the quotient.

#solutionbox[
Answers will vary.
]


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “divide a polynomial by a monomial,” and “divide a polynomial by a binomial.” The rest of the cells are blank.], alt: "This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “divide a polynomial by a monomial,” and “divide a polynomial by a binomial.” The rest of the cells are blank.", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all goals?