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== 10.7#h(0.6em)Polynomials and Factoring


    In Laws of Exponents, we used the first law of exponents to multiply two or more monomials. In this section, we review techniques for multiplying and factoring polynomials of several terms.


    
      
=== Polynomials


      A #strong[polynomial] is a sum of terms in which all the exponents on the variables are whole numbers and no variables appear in the denominator or under a radical. The expressions


      #math.equation(block: true, alt: "0.1 R to the power 4 , d squared plus 32 d minus 21 , and 128 x cubed minus 960 x squared plus 8000")[$0.1 R^(4) , " " " " d^(2) + 32 d − 21 , " " " " " and " " " " " 128 x^(3) − 960 x^(2) + 8000$]
      are all examples of polynomials in one variable.


      An algebraic expression consisting of one term of the form #math.equation(block: false, alt: "c x to the power n")[$c x^(n)$], where #math.equation(block: false, alt: "c")[$c$] is a constant and #math.equation(block: false, alt: "n")[$n$] is a whole number, is called a #strong[monomial]. For example,


      #math.equation(block: true, alt: "y 3 , minus 3 x to the power 8 , and 0.1 R to the power 4")[$y 3 , " " " " − 3 x^(8) , " " " " " and " " " " " 0.1 R^(4)$]
      are monomials. A polynomial is just a sum of one or more monomials.


      A polynomial with exactly two terms, such as #math.equation(block: false, alt: "the fraction 1 over 2 n squared plus the fraction 1 over 2 n")[$display(frac(1, 2)) n^(2) + display(frac(1, 2)) n$], is called a #strong[binomial]. A polynomial with exactly three terms, such as #math.equation(block: false, alt: "d squared plus 32 d minus 21")[$d^(2) + 32 d − 21$] or #math.equation(block: false, alt: "128 x cubed minus 960 x squared plus 8000")[$128 x^(3) − 960 x^(2) + 8000$], is called a #strong[trinomial]. We have no special names for polynomials with more than three terms.


      #examplebox("Example 1")[][
Which of the following expressions are polynomials?


          + #math.equation(block: false, alt: "π r squared")[$π r^(2)$]
+ #math.equation(block: false, alt: "23.4 s to the power 6 minus 47.9 s to the power 4")[$23.4 s^(6) − 47.9 s^(4)$]
+ #math.equation(block: false, alt: "the fraction 2 over 3 w cubed minus the fraction 7 over 3 w squared plus the fraction 1 over 3 w")[$display(frac(2, 3)) w^(3) − display(frac(7, 3)) w^(2) + display(frac(1, 3)) w$]
+ #math.equation(block: false, alt: "7 plus m to the power minus 2")[$7 + m^(− 2)$]
+ #math.equation(block: false, alt: "the fraction x minus 2 over x plus 2")[$display(frac(x − 2, x + 2))$]
+ #math.equation(block: false, alt: "the cube root of 4 y")[$root(3, 4 y)$]


        
        #solutionbox[
The first three are all polynomials. In fact, (a) is a monomial, (b) is a binomial, and (c) is a trinomial. The last three are not polynomials. The variable in (d) has a negative exponent, the variable in (e) occurs in the denominator, and the variable in (f) occurs under a radical.
]
]


      In a polynomial containing only one variable, the greatest exponent that appears on the variable is called the #strong[degree] of the polynomial. If there is no variable at all, then the polynomial is called a constant, and the degree of a constant is zero.


      #examplebox("Example 2")[][
Give the degree of each polynomial.


          + #math.equation(block: false, alt: "b cubed minus 3 b squared plus 3 b minus 1")[$b^(3) − 3 b^(2) + 3 b − 1$]
+ #math.equation(block: false, alt: "10 to the power 10")[$10^(10)$]
+ #math.equation(block: false, alt: "minus 4 w cubed")[$− 4 w^(3)$]
+ #math.equation(block: false, alt: "s squared minus s to the power 6")[$s^(2) − s^(6)$]


        
        #solutionbox[
+ This is a polynomial in the variable #math.equation(block: false, alt: "b")[$b$], and because the greatest exponent on #math.equation(block: false, alt: "b")[$b$] is #math.equation(block: false, alt: "3")[$3$], the degree of this polynomial is #math.equation(block: false, alt: "3")[$3$].
+ This is a constant polynomial, so its degree is #math.equation(block: false, alt: "0")[$0$]. (The exponent on a constant does not affect the degree.)
+ This monomial has degree #math.equation(block: false, alt: "3")[$3$].
+ This is a binomial of degree #math.equation(block: false, alt: "6")[$6$].
]
]


      We can evaluate a polynomial just as we evaluate any other algebraic expression: We replace the variable with a number and simplify the result.


      #examplebox("Example 3")[][
Let #math.equation(block: false, alt: "p open parenthesis x close parenthesis equals minus 2 x squared plus 3 x minus 1")[$p ( x ) = − 2 x^(2) + 3 x − 1$]. Evaluate each of the following.


          + #math.equation(block: false, alt: "p open parenthesis 2 close parenthesis")[$p ( 2 )$]
+ #math.equation(block: false, alt: "p open parenthesis minus 1 close parenthesis")[$p ( − 1 )$]
+ #math.equation(block: false, alt: "p open parenthesis t close parenthesis")[$p ( t )$]
+ #math.equation(block: false, alt: "p open parenthesis t plus 3 close parenthesis")[$p ( t + 3 )$]


        
        #solutionbox[
In each case, we replace #math.equation(block: false, alt: "x")[$x$] by the given value.


          + #math.equation(block: false, alt: "p open parenthesis 2 close parenthesis equals minus 2 open parenthesis 2 close parenthesis squared plus 3 open parenthesis 2 close parenthesis minus 1 equals minus 8 plus 6 minus 1 equals minus 3")[$p ( 2 ) = − 2 ( 2 )^(2) + 3 ( 2 ) − 1 = − 8 + 6 − 1 = − 3$]
+ #math.equation(block: false, alt: "p open parenthesis minus 1 close parenthesis equals minus 2 open parenthesis minus 1 close parenthesis squared plus 3 open parenthesis minus 1 close parenthesis minus 1 equals minus 2 plus open parenthesis minus 3 close parenthesis minus 1 equals minus 6")[$p ( − 1 ) = − 2 ( − 1 )^(2) + 3 ( − 1 ) − 1 = − 2 + ( − 3 ) − 1 = − 6$]
+ #math.equation(block: false, alt: "p open parenthesis t close parenthesis equals minus 2 open parenthesis t close parenthesis squared plus 3 open parenthesis t close parenthesis minus 1 equals minus minus 2 t squared plus 3 t minus 1")[$p ( bold(italic(t)) ) = − 2 ( bold(italic(t)) )^(2) + 3 ( bold(italic(t)) ) − 1 = − − 2 t^(2) + 3 t − 1$]
+ #math.equation(block: false, alt: "p open parenthesis t plus 3 close parenthesis, equals minus 2 open parenthesis t plus 3 close parenthesis squared plus 3 open parenthesis t plus 3 close parenthesis minus 1; equals minus 2 open parenthesis t squared plus 6 t plus 9 close parenthesis plus 3 open parenthesis t plus 3 close parenthesis minus 1; equals minus 2 t squared minus 9 t minus 10")[$p ( bold(italic(t)) + 3 ) & = − 2 ( bold(italic(t)) + 3 )^(2) + 3 ( bold(italic(t)) + 3 ) − 1 \  & = − 2 ( t^(2) + 6 t + 9 ) + 3 ( t + 3 ) − 1 \  & = − 2 t^(2) − 9 t − 10$]
]
]


    
    
      
=== Products of Polynomials


      To multiply polynomials, we use a generalized form of the distributive property:


      #math.equation(block: true, alt: "a open parenthesis b plus c plus d plus ⋯ close parenthesis equals a b plus a c plus a d plus ⋯")[$a ( b + c + d + ⋯ ) = a b + a c + a d + ⋯$]
      To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.


      #examplebox("Example 4")[][
+ #math.equation(block: false, alt: "3 x open parenthesis x plus y plus z close parenthesis, equals 3 x open parenthesis x close parenthesis plus 3 x open parenthesis y close parenthesis plus 3 x open parenthesis z close parenthesis; equals 3 x squared plus 3 x y plus 3 x z")[$3 x ( x + y + z ) & = 3 x ( x ) + 3 x ( y ) + 3 x ( z ) \  & = 3 x^(2) + 3 x y + 3 x z$]
+ #math.equation(block: false, alt: "minus 2 a b squared open parenthesis 3 a squared minus a b plus 2 b squared close parenthesis, equals minus 2 a b squared open parenthesis 3 a squared close parenthesis minus 2 a b squared open parenthesis minus a b close parenthesis minus 2 a b squared open parenthesis 2 b squared close parenthesis; equals minus 6 a cubed b squared plus 2 a squared b cubed minus 4 a b to the power 4")[$− 2 a b^(2) ( 3 a^(2) − a b + 2 b^(2) ) & = − 2 a b^(2) ( 3 a^(2) ) − 2 a b^(2) ( − a b ) − 2 a b^(2) ( 2 b^(2) ) \  & = − 6 a^(3) b^(2) + 2 a^(2) b^(3) − 4 a b^(4)$]
]


    
    
      
=== Products of Binomials


      Products of binomials occur so frequently that it is worthwhile to learn a shortcut for this type of multiplication. We can use the following scheme to perform the multiplication mentally. (See Figure.)


      #figure(figph[figure identifying thefirst, outer, inner, and last terms in a binomial product], alt: "figure identifying thefirst, outer, inner, and last terms in a binomial product", caption: none)


      This process is sometimes called the #strong[FOIL] method, where FOIL represents


      #math.equation(block: true, alt: "the product of the F irst terms; the product of the O uter terms; the product of the I nner terms; the product of the L ast terms")[$& "the product of the" " " bold(italic(F)) "irst terms" \  & "the product of the" " " bold(italic(O)) "uter terms" \  & "the product of the" " " bold(italic(I)) "nner terms" \  & "the product of the" " " bold(italic(L)) "ast terms"$]
      #examplebox("Example 5")[][
#math.equation(block: true, alt: "open parenthesis 2 x minus 1 close parenthesis open parenthesis x plus 3 close parenthesis, equals 2 x squared plus 6 x minus x minus 3; equals 2 x squared plus 5 x minus 3")[$( 2 x − 1 ) ( x + 3 ) & = 2 x^(2) + 6 x − x − 3 \  & = 2 x^(2) + 5 x − 3$]
]


    
    
      
=== Factoring


      We sometimes find it useful to write a polynomial as a single #emph[term] composed of two or more #emph[factors]. This process is the reverse of multiplication and is called #strong[factoring]. For example, observe that


      #math.equation(block: true, alt: "3 x squared plus 6 x equals 3 x open parenthesis x plus 2 close parenthesis")[$3 x^(2) + 6 x = 3 x ( x + 2 )$]
      We will only consider factorization in which the factors have integer coefficients.


    
    
      
=== Common Factors


      We can factor a common factor from a polynomial by using the distributive property in the form


      #math.equation(block: true, alt: "a b plus a c equals a open parenthesis b plus c close parenthesis")[$a b + a c = a ( b + c )$]
      We first identify the common factor. For example, each term of the polynomial


      #math.equation(block: true, alt: "6 x cubed plus 9 x squared minus 3 x")[$6 x^(3) + 9 x^(2) − 3 x$]
      contains the monomial #math.equation(block: false, alt: "3 x")[$3 x$] as a factor; therefore,


      #math.equation(block: true, alt: "6 x cubed plus 9 x squared minus 3 x equals 3 x open parenthesis bar close parenthesis")[$6 x^(3) + 9 x^(2) − 3 x = 3 x ( underline(#h(7.5em)) )$]
      Next, we insert the proper polynomial factor within the parentheses. This factor can be determined by inspection. We ask ourselves for monomials that, when multiplied by #math.equation(block: false, alt: "3 x")[$3 x$], yield #math.equation(block: false, alt: "6 x cubed")[$6 x^(3)$], #math.equation(block: false, alt: "9 x squared")[$9 x^(2)$], and #math.equation(block: false, alt: "minus 3 x")[$− 3 x$], respectively, and obtain


      #math.equation(block: true, alt: "6 x cubed plus 9 x squared minus 3 x equals 3 x open parenthesis 2 x squared plus 3 x minus 1 close parenthesis")[$6 x^(3) + 9 x^(2) − 3 x = 3 x ( 2 x^(2) + 3 x − 1 )$]
      We can check the result of factoring an expression by multiplying the factors. In the example above,


      #math.equation(block: true, alt: "3 x open parenthesis 2 x squared plus 3 x minus 1 close parenthesis equals 6 x cubed plus 9 x squared minus 3 x")[$3 x ( 2 x^(2) + 3 x − 1 ) = 6 x^(3) + 9 x^(2) − 3 x$]
      #examplebox("Example 6")[][
+ #math.equation(block: false, alt: "18 x squared y minus 24 x y squared, equals 6 x y open parenthesis ? minus ? close parenthesis; equals 6 x y open parenthesis 3 x minus 4 y close parenthesis")[$18 x^(2) y − 24 x y^(2) & = 6 x y ( "?" − "?" ) \  & = 6 x y ( 3 x − 4 y )$] #linebreak() because
            #math.equation(block: true, alt: "6 x y open parenthesis 3 x minus 4 y close parenthesis equals 18 x squared y minus 24 x y squared")[$6 x y ( 3 x − 4 y ) = 18 x^(2) y − 24 x y^(2)$]
+ #math.equation(block: false, alt: "y open parenthesis x minus 2 close parenthesis plus z open parenthesis x minus 2 close parenthesis, equals open parenthesis x minus 2 close parenthesis open parenthesis ? minus ? close parenthesis; equals open parenthesis x minus 2 close parenthesis open parenthesis y plus z close parenthesis")[$y ( x − 2 ) + z ( x − 2 ) & = ( x − 2 ) ( "?" − "?" ) \  & = ( x − 2 ) ( y + z )$] #linebreak() because
            #math.equation(block: true, alt: "open parenthesis x minus 2 close parenthesis open parenthesis y plus z close parenthesis equals y open parenthesis x minus 2 close parenthesis plus z open parenthesis x minus 2 close parenthesis")[$( x − 2 ) ( y + z ) = y ( x − 2 ) + z ( x − 2 )$]
]


    
    
      
=== Opposite of a Binomial


      It is often useful to factor #math.equation(block: false, alt: "minus 1")[$− 1$] from the terms of a binomial.


      #math.equation(block: true, alt: "a minus b, equals open parenthesis minus 1 close parenthesis open parenthesis minus a plus b close parenthesis; equals open parenthesis minus 1 close parenthesis open parenthesis b minus a close parenthesis equals minus open parenthesis b minus a close parenthesis")[$a − b & = ( − 1 ) ( − a + b ) \  & = ( − 1 ) ( b − a ) = − ( b − a )$]
      Hence, we have the following important relationship.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Opposite of a Binomial]


        #math.equation(block: true, alt: "a minus b equals minus open parenthesis b minus a close parenthesis")[$a − b = − ( b − a )$]
]


      That is, #math.equation(block: false, alt: "a minus b")[$a − b$] and #math.equation(block: false, alt: "b minus a")[$b − a$] are opposites or negatives of each other.


      #examplebox("Example 7")[][
+ #math.equation(block: false, alt: "3 x minus y equals minus open parenthesis y minus 3 x close parenthesis")[$3 x − y = − ( y − 3 x )$]
+ #math.equation(block: false, alt: "a minus 2 b equals minus open parenthesis 2 b minus a close parenthesis")[$a − 2 b = − ( 2 b − a )$]
]


    
    
      
=== Polynomial Division


      We can divide one polynomial by a polynomial of lesser degree. The quotient will be the sum of a polynomial and a simpler algebraic fraction.


      If the divisor is a monomial, we can simply divide the monomial into each term of the numerator.


      #examplebox("Example 8")[][
Divide #math.equation(block: false, alt: "the fraction 9 x cubed minus 6 x squared plus 4 over 3 x")[$display(frac(9 x^(3) − 6 x^(2) + 4, 3 x))$]


        
        #solutionbox[
Divide #math.equation(block: false, alt: "3 x")[$3 x$] into each term of the numerator.


          #math.equation(block: true, alt: "the fraction 9 x cubed minus 6 x squared plus 4 over 3 x, equals the fraction 9 x cubed over 3 x minus the fraction 6 x squared over 3 x plus the fraction 4 over 3 x; equals 3 x squared minus 2 x plus the fraction 4 over 3 x")[$frac(9 x^(3) − 6 x^(2) + 4, 3 x) & = frac(9 x^(3), 3 x) − frac(6 x^(2), 3 x) + frac(4, 3 x) \  & = 3 x^(2) − 2 x + frac(4, 3 x)$]
          The quotient is the sum of a polynomial, #math.equation(block: false, alt: "3 x squared minus 2 x")[$3 x^(2) − 2 x$], and an algebraic fraction, #math.equation(block: false, alt: "the fraction 4 over 3 x")[$display(frac(4, 3 x))$].
]
]


      If the denominator is not a monomial, we can use a method similar to the long division algorithm used in arithmetic.


      #examplebox("Example 9")[][
Divide #math.equation(block: false, alt: "the fraction 2 x squared plus x minus 7 over x plus 3")[$" " " " display(frac(2 x^(2) + x − 7, x + 3))$]


        
        #solutionbox[
We first write the quotient as a division problem:


          #math.equation(block: true, alt: "x plus 32 x squared plus x minus 7")[$x + 3 #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$2 x^(2) + x − 7$] #h(-0.1em)$]
          and divide #math.equation(block: false, alt: "2 x squared")[$2 x^(2)$] (the first term of the numerator) by #math.equation(block: false, alt: "x")[$x$] (the first term of the denominator) to obtain #math.equation(block: false, alt: "2 x")[$2 x$]. (It may be helpful to write down the division: #math.equation(block: false, alt: "the fraction 2 x squared over 2 x equals x")[$display(frac(2 x^(2), 2 x)) = x$].) We write #math.equation(block: false, alt: "2 x")[$2 bold(italic(x))$] above the quotient bar as the first term of the quotient, as shown below.


          Next, we multiply #math.equation(block: false, alt: "x plus 3")[$x + 3$] by #math.equation(block: false, alt: "2 x")[$2 x$] to obtain #math.equation(block: false, alt: "2 x squared plus 6 x")[$2 x^(2) + 6 x$], and subtract this product from #math.equation(block: false, alt: "2 x squared plus x minus 7")[$2 x^(2) + x − 7$]:


          #math.equation(block: true, alt: "2 x; x plus 32 x squared plus x minus 7, Multiply 2 x by x plus 3.; minus open parenthesis 2 x squared plus 6 x close parenthesis bar, Subtract the result .; minus 5 x minus 7")[$2 bold(italic(x)) #hide($1 + x − 7$) &  &  \ x + 3 #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$2 x^(2) + x − 7$] #h(-0.1em) &  & "Multiply " 2 bold(italic(x)) " by " bold(italic(x)) + 3. \ underline(− ( 2 x^(2) + 6 x ) #hide($11$)) &  & "Subtract the result" . \ − 5 x − 7$]
          Repeating the process, we divide #math.equation(block: false, alt: "minus 5 x")[$− 5 x$] by #math.equation(block: false, alt: "x")[$x$] to obtain #math.equation(block: false, alt: "minus 5")[$− 5$]. We write #math.equation(block: false, alt: "minus 5")[$− 5$] as the second term of the quotient. Then we multiply #math.equation(block: false, alt: "x plus 3")[$x + 3$] by #math.equation(block: false, alt: "minus 5")[$− 5$] to obtain #math.equation(block: false, alt: "minus 5 x minus 15")[$− 5 x − 15$], and subtract:


          #math.equation(block: true, alt: "2 x minus 5; x plus 32 x squared plus x minus 7; minus open parenthesis 2 x squared plus 6 x close parenthesis bar; minus 5 x minus 7, Multiply minus 5 by x plus 3.; minus open parenthesis minus 5 x minus 15 close parenthesis bar, Subtract the result .; 8")[$2 x − 5 #hide($x − 11$) &  &  \ x + 3 #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$2 x^(2) + x − 7$] #h(-0.1em) #hide($11$) &  &  \ underline(− ( 2 x^(2) + 6 x ) #hide($11$)) #hide($11$) &  &  \ − 5 x − 7 #hide($11$) &  & "Multiply " − 5 " by " bold(italic(x)) + 3. \ underline(− ( − 5 x − 15 )) &  & "Subtract the result" . \ 8 #hide($11$)$]
          Because the degree of the remainder, 8, is less than the degree of #math.equation(block: false, alt: "x plus 3")[$x + 3$], the division is finished. The quotient is #math.equation(block: false, alt: "2 x minus 5")[$2 x − 5$], with a remainder of #math.equation(block: false, alt: "8")[$8$]. We write the remainder as a fraction to obtain


          #math.equation(block: true, alt: "the fraction 2 x squared plus x minus 7 over x plus 3 equals 2 x minus 5 plus the fraction 8 over x plus 3")[$display(frac(2 x^(2) + x − 7, x + 3)) = 2 x − 5 + display(frac(8, x + 3))$]
]
]


      When using polynomial division, it helps to write the polynomials in descending powers of the variable. If the numerator is missing any terms, we can insert terms with zero coefficients so that like powers will be aligned. For example, to perform the division


      #math.equation(block: true, alt: "the fraction 3 x minus 1 plus 4 x cubed over 2 x minus 1")[$frac(3 x − 1 + 4 x^(3), 2 x − 1)$]
      we first write the numerator in descending powers as #math.equation(block: false, alt: "4 x cubed plus 3 x minus 1")[$4 x^(3) + 3 x − 1$]. We then insert #math.equation(block: false, alt: "0 x squared")[$0 x^(2)$] between #math.equation(block: false, alt: "4 x cubed")[$4 x^(3)$] and #math.equation(block: false, alt: "3 x")[$3 x$] and set up the quotient as


      #figure(figph[long division box], alt: "long division box", caption: none)


      We then proceed as in Example. You can check that the quotient is


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


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Polynomial
- Common factor
- Degree
- Constant
- Trinomial
- Monomial
- Binomial


      
      
        
==== SKILLS


        Practice each skill in the exercises listed.


        + Identify polynomials: \#1–12
+ Evaluate polynomials: \#13–20
+ Multiply polynomials: \#21–42
+ Factor out a common factor: \#43–68
+ Divide polynomials: \#69–80


      
    
    
      
=== Exercises A.7


      For Problems 1-8, identify the polynomial as a monomial, a binomial, or a trinomial. Give the degree of the polynomial.


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


        
          Binomial; 3


        
      
      
        #math.equation(block: true, alt: "x squared minus 2 x plus 1")[$x^(2) − 2 x + 1$]


      
      
        #math.equation(block: true, alt: "5 n to the power 4")[$5 n^(4)$]


        
          Monomial; 4


        
      
      
        #math.equation(block: true, alt: "3 n plus 1")[$3 n + 1$]


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


        
          Trinomial; 2


        
      
      
        #math.equation(block: true, alt: "r cubed")[$r^(3)$]


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


        
          Trinomial; 3


        
      
      
        #math.equation(block: true, alt: "3 y squared plus 1")[$3 y^(2) + 1$]


      
      Which of the expressions in Problems 9-12 are not polynomials?


      
        + #math.equation(block: false, alt: "1 minus 0.04 t squared")[$1 − 0.04 t^(2)$]
+ #math.equation(block: false, alt: "3 x squared minus 4 x plus the fraction 2 over x")[$3 x^(2) − 4 x + display(frac(2, x))$]
+ #math.equation(block: false, alt: "2 the square root of z minus 7 z cubed plus 2")[$2 sqrt(z) − 7 z^(3) + 2$]
+ #math.equation(block: false, alt: "the square root of 2 w cubed plus the fraction 3 over 4 w squared minus w")[$sqrt(2) w^(3) + display(frac(3, 4)) w^(2) − w$]


        
          b and c


        
      
      
        + #math.equation(block: false, alt: "the square root of 3 p squared minus 7 p plus 2")[$sqrt(3) p^(2) − 7 p + 2$]
+ #math.equation(block: false, alt: "2 h to the power 4 / 3 plus 6 h to the power 1 / 3 minus 2")[$2 h^(4 / 3) + 6 h^(1 / 3) − 2$]
+ #math.equation(block: false, alt: "the fraction 2 over x squared minus 6 x plus 5")[$display(frac(2, x^(2) − 6 x + 5))$]
+ #math.equation(block: false, alt: "the fraction 1 over 4 y to the power minus 2 plus 3 y to the power minus 1 plus 4")[$display(frac(1, 4)) y^(− 2) + 3 y^(− 1) + 4$]


      
      
        + #math.equation(block: false, alt: "the fraction 1 over m squared plus 3")[$display(frac(1, m^(2) + 3))$]
+ #math.equation(block: false, alt: "v squared minus 16 plus 2 to the power v")[$v^(2) − 16 + 2^(v)$]
+ #math.equation(block: false, alt: "the square root of x cubed minus 4 x")[$sqrt(x^(3) − 4 x)$]
+ #math.equation(block: false, alt: "the fraction m to the power 4 over 12")[$display(frac(m^(4), 12))$]


        
          a, b, c


        
      
      
        + #math.equation(block: false, alt: "3 to the power t minus 5 t cubed plus 2")[$3^(t) − 5 t^(3) + 2$]
+ #math.equation(block: false, alt: "the fraction q plus 3 over q minus 1")[$display(frac(q + 3, q − 1))$]
+ #math.equation(block: false, alt: "c to the power 1 / 2 minus c")[$c^(1 / 2) − c$]
+ #math.equation(block: false, alt: "the cube root of d plus 1")[$root(3, d + 1)$]


      
      For Problems 13-20, evaluate the polynomial function for the given values of the variable.


      
        #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed minus 3 x squared plus x plus 1")[$P ( x ) = x^(3) − 3 x^(2) + x + 1$]


        + #math.equation(block: false, alt: "x equals 2")[$x = 2$]
+ #math.equation(block: false, alt: "x equals minus 2")[$x = − 2$]
+ #math.equation(block: false, alt: "x equals 2 b")[$x = 2 b$]


        
          + #math.equation(block: false, alt: "minus 1")[$− 1$]
+ #math.equation(block: false, alt: "minus 21")[$− 21$]
+ #math.equation(block: false, alt: "8 b cubed minus 12 b squared plus 2 b plus 1")[$8 b^(3) − 12 b^(2) + 2 b + 1$]


        
      
      
        #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals 2 x cubed plus x squared minus 3 x plus 4")[$P ( x ) = 2 x^(3) + x^(2) − 3 x + 4$]


        + #math.equation(block: false, alt: "x equals 3")[$x = 3$]
+ #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]
+ #math.equation(block: false, alt: "x equals minus a")[$x = − a$]


      
      
        #math.equation(block: true, alt: "Q open parenthesis t close parenthesis equals t squared plus 3 t plus 1")[$Q ( t ) = t^(2) + 3 t + 1$]


        + #math.equation(block: false, alt: "t equals the fraction 1 over 2")[$t = display(frac(1, 2))$]
+ #math.equation(block: false, alt: "t equals minus the fraction 1 over 3")[$t = − display(frac(1, 3))$]
+ #math.equation(block: false, alt: "t equals minus w")[$t = − w$]


        
          + #math.equation(block: false, alt: "the fraction 11 over 4")[$display(frac(11, 4))$]
+ #math.equation(block: false, alt: "the fraction 1 over 9")[$display(frac(1, 9))$]
+ #math.equation(block: false, alt: "w squared minus 3 w plus 1")[$w^(2) − 3 w + 1$]


        
      
      
        #math.equation(block: true, alt: "Q open parenthesis t close parenthesis equals 2 t squared minus t plus 1")[$Q ( t ) = 2 t^(2) − t + 1$]


        + #math.equation(block: false, alt: "t equals the fraction 1 over 4")[$t = display(frac(1, 4))$]
+ #math.equation(block: false, alt: "t equals minus the fraction 1 over 2")[$t = − display(frac(1, 2))$]
+ #math.equation(block: false, alt: "t equals 3 v")[$t = 3 v$]


      
      
        #math.equation(block: true, alt: "R open parenthesis z close parenthesis equals 3 z to the power 4 minus 2 z squared plus 3")[$R ( z ) = 3 z^(4) − 2 z^(2) + 3$]


        + #math.equation(block: false, alt: "z equals 1.8")[$z = 1.8$]
+ #math.equation(block: false, alt: "z equals minus 2.6")[$z = − 2.6$]
+ #math.equation(block: false, alt: "z equals k minus 1")[$z = k − 1$]


        
          + #math.equation(block: false, alt: "28.0128")[$28.0128$]
+ #math.equation(block: false, alt: "126.5728")[$126.5728$]
+ #math.equation(block: false, alt: "3 k to the power 4 minus 12 k cubed plus 16 k squared minus 8 k plus 4")[$3 k^(4) − 12 k^(3) + 16 k^(2) − 8 k + 4$]


        
      
      
        #math.equation(block: true, alt: "R open parenthesis z close parenthesis equals z to the power 4 plus 4 z minus 2")[$R ( z ) = z^(4) + 4 z − 2$]


        + #math.equation(block: false, alt: "z equals 2.1")[$z = 2.1$]
+ #math.equation(block: false, alt: "z equals minus 3.1")[$z = − 3.1$]
+ #math.equation(block: false, alt: "z equals h plus 2")[$z = h + 2$]


      
      
        #math.equation(block: true, alt: "N open parenthesis a close parenthesis equals a to the power 6 minus a to the power 5")[$N ( a ) = a^(6) − a^(5)$]


        + #math.equation(block: false, alt: "a equals minus 1")[$a = − 1$]
+ #math.equation(block: false, alt: "a equals minus 2")[$a = − 2$]
+ #math.equation(block: false, alt: "a equals the fraction m over 3")[$a = display(frac(m, 3))$]


        
          + #math.equation(block: false, alt: "2")[$2$]
+ #math.equation(block: false, alt: "96")[$96$]
+ #math.equation(block: false, alt: "the fraction m to the power 6 over 729 minus the fraction m to the power 5 over 243")[$display(frac(m^(6), 729)) − display(frac(m^(5), 243))$]


        
      
      
        #math.equation(block: true, alt: "N open parenthesis a close parenthesis equals a to the power 5 minus a to the power 4")[$N ( a ) = a^(5) − a^(4)$]


        + #math.equation(block: false, alt: "a equals minus 1")[$a = − 1$]
+ #math.equation(block: false, alt: "a equals minus 2")[$a = − 2$]
+ #math.equation(block: false, alt: "a equals the fraction q over 2")[$a = display(frac(q, 2))$]


      
      For Problems 21-42, write the product as a polynomial and simplify.


      
        #math.equation(block: true, alt: "4 y open parenthesis x minus 2 y close parenthesis")[$4 y ( x − 2 y )$]


        
          #math.equation(block: true, alt: "4 x y minus 8 y squared")[$4 x y − 8 y^(2)$]


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


      
      
        #math.equation(block: true, alt: "minus 6 x open parenthesis 2 x squared minus x plus 1 close parenthesis")[$− 6 x ( 2 x^(2) − x + 1 )$]


        
          #math.equation(block: true, alt: "minus 12 x cubed plus 6 x squared minus 6 x")[$− 12 x^(3) + 6 x^(2) − 6 x$]


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


      
      
        #math.equation(block: true, alt: "a squared b open parenthesis 3 a squared minus 2 a b minus b close parenthesis")[$a^(2) b ( 3 a^(2) − 2 a b − b )$]


        
          #math.equation(block: true, alt: "3 a to the power 4 b minus 2 a cubed b squared minus a squared b squared")[$3 a^(4) b − 2 a^(3) b^(2) − a^(2) b^(2)$]


        
      
      
        #math.equation(block: true, alt: "a b cubed open parenthesis minus a squared b squared plus 4 a b minus 3 close parenthesis")[$a b^(3) ( − a^(2) b^(2) + 4 a b − 3 )$]


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


        
          #math.equation(block: true, alt: "8 x cubed y to the power 7 minus 4 x cubed y to the power 4 minus 6 x to the power 5 y to the power 5")[$8 x^(3) y^(7) − 4 x^(3) y^(4) − 6 x^(5) y^(5)$]


        
      
      
        #math.equation(block: true, alt: "5 x squared y squared open parenthesis 3 x to the power 4 y squared plus 3 x squared y minus x y to the power 6 close parenthesis")[$5 x^(2) y^(2) ( 3 x^(4) y^(2) + 3 x^(2) y − x y^(6) )$]


      
      
        #math.equation(block: true, alt: "open parenthesis n plus 2 close parenthesis open parenthesis n plus 8 close parenthesis")[$( n + 2 ) ( n + 8 )$]


        
          #math.equation(block: true, alt: "n squared plus 10 n plus 16")[$n^(2) + 10 n + 16$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis r minus 1 close parenthesis open parenthesis r minus 6 close parenthesis")[$( r − 1 ) ( r − 6 )$]


      
      
        #math.equation(block: true, alt: "open parenthesis r plus 5 close parenthesis open parenthesis r minus 2 close parenthesis")[$( r + 5 ) ( r − 2 )$]


        
          #math.equation(block: true, alt: "r squared plus 3 r minus 10")[$r^(2) + 3 r − 10$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis z minus 3 close parenthesis open parenthesis z plus 5 close parenthesis")[$( z − 3 ) ( z + 5 )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 2 z plus 1 close parenthesis open parenthesis z minus 3 close parenthesis")[$( 2 z + 1 ) ( z − 3 )$]


        
          #math.equation(block: true, alt: "2 z squared minus 5 z minus 3")[$2 z^(2) − 5 z − 3$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 3 t minus 1 close parenthesis open parenthesis 2 t plus 1 close parenthesis")[$( 3 t − 1 ) ( 2 t + 1 )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 4 r plus 3 s close parenthesis open parenthesis 2 r minus s close parenthesis")[$( 4 r + 3 s ) ( 2 r − s )$]


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


        
      
      
        #math.equation(block: true, alt: "open parenthesis 2 z minus w close parenthesis open parenthesis 3 z plus 5 w close parenthesis")[$( 2 z − w ) ( 3 z + 5 w )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 2 x minus 3 y close parenthesis open parenthesis 3 x minus 2 y close parenthesis")[$( 2 x − 3 y ) ( 3 x − 2 y )$]


        
          #math.equation(block: true, alt: "6 x squared minus 13 x y plus 6 y squared")[$6 x^(2) − 13 x y + 6 y^(2)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 3 a plus 5 b close parenthesis open parenthesis 3 a plus 4 b close parenthesis")[$( 3 a + 5 b ) ( 3 a + 4 b )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 3 t minus 4 s close parenthesis open parenthesis 3 t plus 4 s close parenthesis")[$( 3 t − 4 s ) ( 3 t + 4 s )$]


        
          #math.equation(block: true, alt: "9 t squared minus 16 s squared")[$9 t^(2) − 16 s^(2)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 2 x minus 3 z close parenthesis open parenthesis 2 x plus 3 z close parenthesis")[$( 2 x − 3 z ) ( 2 x + 3 z )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 2 a squared plus b squared close parenthesis open parenthesis a squared minus 3 b squared close parenthesis")[$( 2 a^(2) + b^(2) ) ( a^(2) − 3 b^(2) )$]


        
          #math.equation(block: true, alt: "2 a to the power 4 minus 5 a squared b squared minus 3 b to the power 4")[$2 a^(4) − 5 a^(2) b^(2) − 3 b^(4)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis s squared minus 5 t squared close parenthesis open parenthesis 3 s squared plus 2 t squared close parenthesis")[$( s^(2) − 5 t^(2) ) ( 3 s^(2) + 2 t^(2) )$]


      
      For Problems 43-60, factor completely. Check your answers by multiplying factors.


      
        #math.equation(block: true, alt: "4 x squared z plus 8 x z")[$4 x^(2) z + 8 x z$]


        
          #math.equation(block: true, alt: "4 x z open parenthesis x plus 2 close parenthesis")[$4 x z ( x + 2 )$]


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


      
      
        #math.equation(block: true, alt: "3 n to the power 4 minus 6 n cubed plus 12 n squared")[$3 n^(4) − 6 n^(3) + 12 n^(2)$]


        
          #math.equation(block: true, alt: "3 n squared open parenthesis n squared minus 2 n plus 4 close parenthesis")[$3 n^(2) ( n^(2) − 2 n + 4 )$]


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


      
      
        #math.equation(block: true, alt: "15 r squared s plus 18 r s squared minus 3 r")[$15 r^(2) s + 18 r s^(2) − 3 r$]


        
          #math.equation(block: true, alt: "3 r open parenthesis 5 r s plus 6 s squared minus 1 close parenthesis")[$3 r ( 5 r s + 6 s^(2) − 1 )$]


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


      
      
        #math.equation(block: true, alt: "3 m squared n to the power 4 minus 6 m cubed n cubed plus 14 m cubed n squared")[$3 m^(2) n^(4) − 6 m^(3) n^(3) + 14 m^(3) n^(2)$]


        
          #math.equation(block: true, alt: "m squared n squared open parenthesis 3 n squared minus 6 m n plus 14 m close parenthesis")[$m^(2) n^(2) ( 3 n^(2) − 6 m n + 14 m )$]


        
      
      
        #math.equation(block: true, alt: "6 x cubed y minus 6 x y cubed plus 12 x squared y squared")[$6 x^(3) y − 6 x y^(3) + 12 x^(2) y^(2)$]


      
      
        #math.equation(block: true, alt: "15 a to the power 4 b cubed c to the power 4 minus 12 a squared b squared c to the power 5 plus 6 a squared b cubed c to the power 4")[$15 a^(4) b^(3) c^(4) − 12 a^(2) b^(2) c^(5) + 6 a^(2) b^(3) c^(4)$]


        
          #math.equation(block: true, alt: "3 a squared b squared c to the power 4 open parenthesis 5 a squared b minus 4 c plus 2 b close parenthesis")[$3 a^(2) b^(2) c^(4) ( 5 a^(2) b − 4 c + 2 b )$]


        
      
      
        #math.equation(block: true, alt: "14 x y to the power 4 z cubed plus 21 x squared y cubed z squared minus 28 x cubed y squared z to the power 5")[$14 x y^(4) z^(3) + 21 x^(2) y^(3) z^(2) − 28 x^(3) y^(2) z^(5)$]


      
      
        #math.equation(block: true, alt: "a open parenthesis a plus 3 close parenthesis plus b open parenthesis a plus 3 close parenthesis")[$a ( a + 3 ) + b ( a + 3 )$]


        
          #math.equation(block: true, alt: "open parenthesis a plus b close parenthesis open parenthesis a plus 3 close parenthesis")[$( a + b ) ( a + 3 )$]


        
      
      
        #math.equation(block: true, alt: "b open parenthesis a minus 2 close parenthesis plus a open parenthesis a minus 2 close parenthesis")[$b ( a − 2 ) + a ( a − 2 )$]


      
      
        #math.equation(block: true, alt: "y open parenthesis y minus 2 close parenthesis minus 3 x open parenthesis y minus 2 close parenthesis")[$y ( y − 2 ) − 3 x ( y − 2 )$]


        
          #math.equation(block: true, alt: "open parenthesis y minus 3 x close parenthesis open parenthesis y minus 2 close parenthesis")[$( y − 3 x ) ( y − 2 )$]


        
      
      
        #math.equation(block: true, alt: "2 x open parenthesis x plus 3 close parenthesis minus y open parenthesis x plus 3 close parenthesis")[$2 x ( x + 3 ) − y ( x + 3 )$]


      
      
        #math.equation(block: true, alt: "4 open parenthesis x minus 2 close parenthesis squared minus 8 x open parenthesis x minus 2 close parenthesis cubed")[$4 ( x − 2 )^(2) − 8 x ( x − 2 )^(3)$]


        
          #math.equation(block: true, alt: "4 open parenthesis x minus 2 close parenthesis squared open parenthesis minus 2 x squared plus 4 x plus 1 close parenthesis")[$4 ( x − 2 )^(2) ( − 2 x^(2) + 4 x + 1 )$]


        
      
      
        #math.equation(block: true, alt: "6 open parenthesis x plus 1 close parenthesis minus 3 x open parenthesis x plus 1 close parenthesis squared")[$6 ( x + 1 ) − 3 x ( x + 1 )^(2)$]


      
      
        #math.equation(block: true, alt: "x open parenthesis x minus 5 close parenthesis squared minus x squared open parenthesis x minus 5 close parenthesis cubed")[$x ( x − 5 )^(2) − x^(2) ( x − 5 )^(3)$]


        
          #math.equation(block: true, alt: "x open parenthesis x minus 5 close parenthesis squared open parenthesis minus x squared plus 5 x plus 1 close parenthesis")[$x ( x − 5 )^(2) ( − x^(2) + 5 x + 1 )$]


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


      
      For Problems 61-68, supply the missing factors or terms.


      
        #math.equation(block: true, alt: "3 m minus n equals minus open parenthesis ? close parenthesis")[$3 m − n = − ( " " " " " " " ? " " " " " " " )$]


        
          #math.equation(block: true, alt: "minus open parenthesis 2 n minus 3 m close parenthesis")[$− ( 2 n − 3 m )$]


        
      
      
        #math.equation(block: true, alt: "2 a minus b equals minus open parenthesis ? close parenthesis")[$2 a − b = − ( " " " " " " " ? " " " " " " " )$]


      
      
        #math.equation(block: true, alt: "minus 2 x plus 2 equals minus 2 open parenthesis ? close parenthesis")[$− 2 x + 2 = − 2 ( " " " " " " " ? " " " " " " " )$]


        
          #math.equation(block: true, alt: "minus 2 open parenthesis x minus 1 close parenthesis")[$− 2 ( x − 1 )$]


        
      
      
        #math.equation(block: true, alt: "minus 6 x minus 9 equals minus 3 open parenthesis ? close parenthesis")[$− 6 x − 9 = − 3 ( " " " " " " " ? " " " " " " " )$]


      
      
        #math.equation(block: true, alt: "minus a b minus a c equals ? open parenthesis b plus c close parenthesis")[$− a b − a c = " ? " ( b + c )$]


        
          #math.equation(block: true, alt: "minus a open parenthesis b plus c close parenthesis")[$− a ( b + c )$]


        
      
      
        #math.equation(block: true, alt: "minus a squared plus a b equals ? open parenthesis a minus b close parenthesis")[$− a^(2) + a b = " ? " ( a − b )$]


      
      
        #math.equation(block: true, alt: "2 x minus y plus 3 z equals minus open parenthesis ? close parenthesis")[$2 x − y + 3 z = − ( " " " " " " " ? " " " " " " " )$]


        
          #math.equation(block: true, alt: "minus open parenthesis minus 2 x plus y minus 3 z close parenthesis")[$− ( − 2 x + y − 3 z )$]


        
      
      
        #math.equation(block: true, alt: "3 x plus 3 y minus 2 z equals minus open parenthesis ? close parenthesis")[$3 x + 3 y − 2 z = − ( " " " " " " " ? " " " " " " " )$]


      
      For Problems 69-80, divide.


      
        #math.equation(block: true, alt: "the fraction 18 r squared s squared minus 15 r s plus 6 over 3 r s")[$display(frac(18 r^(2) s^(2) − 15 r s + 6, 3 r s))$]


        
          #math.equation(block: true, alt: "6 r s minus 5 plus the fraction 2 over r s")[$6 r s − 5 + display(frac(2, r s))$]


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


      
      
        #math.equation(block: true, alt: "the fraction 15 s to the power 10 minus 21 s to the power 5 plus 6 over minus 3 s squared")[$display(frac(15 s^(10) − 21 s^(5) + 6, − 3 s^(2)))$]


        
          #math.equation(block: true, alt: "minus 5 s to the power 8 plus 7 s cubed minus the fraction 2 over s squared")[$− 5 s^(8) + 7 s^(3) − display(frac(2, s^(2)))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 25 m to the power 6 minus 15 m to the power 4 plus 7 over minus 5 m cubed")[$display(frac(25 m^(6) − 15 m^(4) + 7, − 5 m^(3)))$]


      
      
        #math.equation(block: true, alt: "the fraction 4 y squared plus 12 y plus 7 over 2 y plus 1")[$display(frac(4 y^(2) + 12 y + 7, 2 y + 1))$]


        
          #math.equation(block: true, alt: "2 y plus 5 plus the fraction 2 over 2 y plus 1")[$2 y + 5 + display(frac(2, 2 y + 1))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 4 t squared minus 4 t minus 5 over 2 t minus 1")[$display(frac(4 t^(2) − 4 t − 5, 2 t − 1))$]


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


        
          #math.equation(block: true, alt: "x squared plus 4 x plus 9 plus the fraction 19 over x minus 2")[$x^(2) + 4 x + 9 + display(frac(19, x − 2))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 2 x cubed minus 3 x squared minus 2 x plus 4 over x plus 1")[$display(frac(2 x^(3) − 3 x^(2) − 2 x + 4, x + 1))$]


      
      
        #math.equation(block: true, alt: "the fraction 4 z squared plus 5 z plus 8 z to the power 4 plus 3 over 2 z plus 1")[$display(frac(4 z^(2) + 5 z + 8 z^(4) + 3, 2 z + 1))$]


        
          #math.equation(block: true, alt: "4 z cubed minus 2 z squared plus 3 z plus 1 plus the fraction 2 over 2 z plus 1")[$4 z^(3) − 2 z^(2) + 3 z + 1 + display(frac(2, 2 z + 1))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 7 minus 3 t cubed minus 23 t squared plus 10 t to the power 4 over 2 t plus 3")[$display(frac(7 − 3 t^(3) − 23 t^(2) + 10 t^(4), 2 t + 3))$]


      
      
        #math.equation(block: true, alt: "the fraction x to the power 4 minus 1 over x minus 2")[$display(frac(x^(4) − 1, x − 2))$]


        
          #math.equation(block: true, alt: "x cubed plus 2 x squared plus 4 x plus 8 plus the fraction 15 over x minus 2")[$x^(3) + 2 x^(2) + 4 x + 8 + display(frac(15, x − 2))$]


        
      
      
        #math.equation(block: true, alt: "the fraction y to the power 5 plus 1 over y minus 1")[$display(frac(y^(5) + 1, y − 1))$]