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== 10.5#h(0.6em)Chapter Summary and Review


    
      
=== Key Concepts


      + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Polar Coordinates]

The #strong[polar coordinates] of a point #math.equation(block: false, alt: "P")[$P$] in the plane are #math.equation(block: false, alt: "open parenthesis r , θ close parenthesis")[$( r , θ )$], where

- #math.equation(block: false, alt: "vertical bar r vertical bar")[$| r |$]  is the #strong[distance] from  to the pole,
- #math.equation(block: false, alt: "θ")[$θ$]   is the #strong[angle] measured counterclockwise from the polar axis to the ray through #math.equation(block: false, alt: "P")[$P$] from the pole.

#figure(figph[polar], alt: "polar", caption: none)
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Non-Uniqueness of Polar Coordinates]

+ Any point with polar coordinates #math.equation(block: false, alt: "open parenthesis r , θ close parenthesis")[$( r , θ )$] also has coordinates #math.equation(block: false, alt: "open parenthesis r , θ plus 2 k π close parenthesis")[$( r , θ + 2 k π )$], where #math.equation(block: false, alt: "k")[$k$] is an integer.
+ The point #math.equation(block: false, alt: "open parenthesis r , θ close parenthesis")[$( r , θ )$] can also be designated by #math.equation(block: false, alt: "open parenthesis minus r , θ plus π close parenthesis")[$( − r , θ + π )$].
+ The pole has coordinates #math.equation(block: false, alt: "open parenthesis 0 , θ close parenthesis")[$( 0 , θ )$], for any value of #math.equation(block: false, alt: "θ")[$θ$].
]
+ In the polar plane, the coordinate grid lines are circles centered at the pole, with equations #math.equation(block: false, alt: "r equals k")[$r = k$], and lines through the pole, with equations #math.equation(block: false, alt: "θ equals k")[$θ = k$].
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Conversion Equations]

+ To convert from polar coordinates #math.equation(block: false, alt: "open parenthesis r , θ close parenthesis")[$( r , θ )$] to Cartesian:
      #math.equation(block: true, alt: "x, equals r cos ⁡ open parenthesis θ close parenthesis; y, equals r sin ⁡ open parenthesis θ close parenthesis")[$x & = r cos ( θ ) \ y & = r sin ( θ )$]
+ To convert from Cartesian coordinates #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] to polar:
      #math.equation(block: true, alt: "r equals the square root of x squared plus y squared; tan ⁡ open parenthesis θ close parenthesis equals the fraction y over x")[$& r = sqrt(x^(2) + y^(2)) \  & tan ( θ ) = display(frac(y, x))$]

where the choice of #math.equation(block: false, alt: "θ")[$θ$] depends on the quadrant.
]
+ To convert an equation from Cartesian to polar coordinates, we replace each #math.equation(block: false, alt: "x")[$x$] with #math.equation(block: false, alt: "r cos ⁡ open parenthesis θ close parenthesis")[$r cos ( θ )$] and each with #math.equation(block: false, alt: "y")[$y$] with #math.equation(block: false, alt: "r sin close parenthesis")[$r sin \)$]. To convert an equation from polar to Cartesian coordinates, look for expressions of the form #math.equation(block: false, alt: "r cos ⁡ open parenthesis θ close parenthesis , r sin ⁡ open parenthesis θ close parenthesis , r squared")[$r cos ( θ ) , " " r sin ( θ ) , " " r^(2)$], or #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$].
+ When graphing an equation in polar coordinates, we think of sweeping around the pole in the counterclockwise direction, and at each angle #math.equation(block: false, alt: "θ")[$θ$] the #math.equation(block: false, alt: "r")[$r$]-value tells us how far the graph is from the pole.
+ Standard graphs in polar coordinates include circles and roses, cardioids and limaçons, lemniscates, and spirals.
+ To find the intersection points of the polar graphs #math.equation(block: false, alt: "r equals f open parenthesis θ close parenthesis")[$r = f ( θ )$] and #math.equation(block: false, alt: "r equals g open parenthesis θ close parenthesis")[$r = g ( θ )$] we solve the equation #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals g open parenthesis θ close parenthesis")[$f ( θ ) = g ( θ )$]. In addition, we should always check whether the pole is a point on both graphs.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Imaginary Unit]

We define the imaginary unit, #strong[#math.equation(block: false, alt: "i")[$i$]],  by

#math.equation(block: true, alt: "i squared equals minus 1 or i equals the square root of minus 1")[$i^(2) = − 1 " " " " " " " " " " " " "or" " " " " " " " " " " " " i = sqrt(− 1)$]
]
+ The square root of a negative number is an imaginary number:  if #math.equation(block: false, alt: "a greater than 0 , the square root of minus a equals i the square root of a")[$a > 0 , " " sqrt(− a) = i sqrt(a)$]
+ A complex number #math.equation(block: false, alt: "z")[$z$] is the sum of a real number and an imaginary number, #math.equation(block: false, alt: "z equals a plus b i")[$z = a + b i$].
+ We can perform the four arithmetic operations on complex numbers.
    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Operations on Complex Numbers]

#math.equation(block: true, alt: "z sub 1 plus z sub 2 equals open parenthesis a plus b i close parenthesis plus open parenthesis c plus d i close parenthesis equals open parenthesis a plus c close parenthesis plus open parenthesis b plus d close parenthesis i")[$z_(1) + z_(2) = ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i$]#math.equation(block: true, alt: "z sub 1 minus z sub 2 equals open parenthesis a plus b i close parenthesis minus open parenthesis c plus d i close parenthesis equals open parenthesis a minus c close parenthesis plus open parenthesis b minus d close parenthesis i")[$z_(1) − z_(2) = ( a + b i ) − ( c + d i ) = ( a − c ) + ( b − d ) i$]#math.equation(block: true, alt: "z sub 1 z sub 2 equals open parenthesis a plus b i close parenthesis open parenthesis c plus d i close parenthesis equals open parenthesis a c minus b d close parenthesis plus open parenthesis a d plus b c close parenthesis i")[$z_(1) z_(2) = ( a + b i ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i$]#math.equation(block: true, alt: "the fraction z sub 1 over z sub 2 equals the fraction a plus b i over c plus d i equals the fraction a plus b i over c plus d i times the fraction c minus d i over c minus d i equals the fraction a c plus b d over c squared plus d squared plus the fraction b c minus a d over c squared plus d squared i")[$display(frac(z_(1), z_(2))) = display(frac(a + b i, c + d i)) = display(frac(a + b i, c + d i)) ⋅ display(frac(c − d i, c − d i)) = display(frac(a c + b d, c^(2) + d^(2))) + display(frac(b c − a d, c^(2) + d^(2))) i$]
]
+ The product of a nonzero complex number and its conjugate is always a positive real number.
      #math.equation(block: true, alt: "z z bar equals open parenthesis a plus b i close parenthesis open parenthesis a minus b i close parenthesis equals a squared minus b squared i squared equals a squared minus b squared open parenthesis minus 1 close parenthesis equals a squared plus b squared")[$z overline(z) = ( a + b i ) ( a − b i ) = a^(2) − b^(2) i^(2) = a^(2) − b^(2) ( − 1 ) = a^(2) + b^(2)$].
+ We can graph complex numbers in the complex plane.
+ We can visualize the sum of two complex numbers by vector addition in the complex plane.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fundamental Theorem of Algebra]

Let #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] be a polynomial of degree #math.equation(block: false, alt: "n greater than or equal to 1")[$n ≥ 1$]. Then #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] has exactly #math.equation(block: false, alt: "n")[$n$] complex zeros.
]
+ The nonreal zeros of a polynomial with real coefficients always occur in conjugate pairs.
+ Multiplying a complex number by #math.equation(block: false, alt: "i")[$i$] rotates its graph by #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] around the origin.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Polar Form for a Complex Number]

The complex number #math.equation(block: false, alt: "z equals a plus b i")[$z = a + b i$] can be written in the polar form

#math.equation(block: true, alt: "z equals r open parenthesis cos ⁡ open parenthesis θ close parenthesis plus i sin ⁡ open parenthesis θ close parenthesis close parenthesis")[$z = r ( cos ( θ ) + i sin ( θ ) )$]where

#math.equation(block: true, alt: "r equals the square root of a squared plus b squared")[$r = sqrt(a^(2) + b^(2))$]and #math.equation(block: false, alt: "θ")[$θ$] is defined by

#math.equation(block: true, alt: "a equals r cos ⁡ open parenthesis θ close parenthesis , b equals r sin ⁡ open parenthesis θ close parenthesis , 0 less than or equal to θ less than or equal to 2 π")[$a = r cos ( θ ) , " " " " " " " " b = r sin ( θ ) , " " " " " " " " 0 ≤ θ ≤ 2 π$]The angle #math.equation(block: false, alt: "θ")[$θ$] is called the #strong[argument] of the complex number, and #math.equation(block: false, alt: "r")[$r$] is its length, or #strong[modulus].
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product and Quotient in Polar Form]

If #math.equation(block: false, alt: "z sub 1 equals r open parenthesis cos ⁡ α plus i sin ⁡ α close parenthesis")[$z_(1) = r ( cos α + i sin α )$] and #math.equation(block: false, alt: "z sub 2 equals R open parenthesis cos ⁡ β plus i sin ⁡ β close parenthesis")[$z_(2) = R ( cos β + i sin β )$], then

#math.equation(block: true, alt: "z sub 1 z sub 2 equals r R open parenthesis cos ⁡ open parenthesis α plus β close parenthesis plus i sin ⁡ open parenthesis α plus β close parenthesis close parenthesis")[$z_(1) z_(2) = r R ( cos ( α + β ) + i sin ( α + β ) )$]and

#math.equation(block: true, alt: "the fraction z sub 1 over z sub 2 equals the fraction r over R open parenthesis cos ⁡ open parenthesis α minus β close parenthesis plus i sin ⁡ open parenthesis α minus β close parenthesis close parenthesis")[$display(frac(z_(1), z_(2))) = display(frac(r, R)) ( cos ( α − β ) + i sin ( α − β ) )$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[De Moivre's Theorem]

If #math.equation(block: false, alt: "z equals r open parenthesis cos ⁡ open parenthesis α close parenthesis plus i sin ⁡ open parenthesis α close parenthesis close parenthesis")[$z = r ( cos ( α ) + i sin ( α ) )$] is a complex number in polar form, and #math.equation(block: false, alt: "n")[$n$] is a positive integer, then

#math.equation(block: true, alt: "z to the power n equals r to the power n open parenthesis cos ⁡ open parenthesis n α close parenthesis plus i sin ⁡ open parenthesis n α close parenthesis close parenthesis")[$z^(n) = r^(n) ( cos ( n α ) + i sin ( n α ) )$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Roots of a Complex Number]

A complex number #math.equation(block: false, alt: "z equals r open parenthesis cos ⁡ open parenthesis α close parenthesis plus i sin ⁡ open parenthesis α close parenthesis close parenthesis")[$z = r ( cos ( α ) + i sin ( α ) )$] in polar form has #math.equation(block: false, alt: "n")[$n$] complex #math.equation(block: false, alt: "n")[$n$]th roots, given by

#math.equation(block: true, alt: "z sub k equals r to the power 1 / n open parenthesis cos ⁡ open parenthesis the fraction α plus 2 π k over n close parenthesis plus i sin ⁡ open parenthesis the fraction α plus 2 π k over n close parenthesis close parenthesis")[$bold(italic(z))_(bold(italic(k))) = bold(italic(r))^(1 / bold(italic(n))) ( bold(cos) ( display(frac(bold(italic(α)) + 2 bold(italic(π)) bold(italic(k)), bold(italic(n)))) ) + bold(italic(i)) bold(sin) ( display(frac(bold(italic(α)) + 2 bold(italic(π)) bold(italic(k)), bold(italic(n)))) ) )$]for #math.equation(block: false, alt: "k equals 0 , 1 , 2 , ⋯ , n minus 1")[$k = 0 , " " 1 , " " 2 , ⋯ , " " n − 1$].
]


    
    
      
=== Review Problems


      For Problems 1–4, use the grid at right to plot the points whose polar coordinates are given.


      #figure(figph[polar grid], alt: "polar grid", caption: none)


      
        #math.equation(block: true, alt: "open parenthesis 4 , the fraction 5 π over 6 close parenthesis")[$( 4 , display(frac(5 π, 6)) )$]


        
          #figure(figph[polar plot], alt: "polar plot", caption: none)


        
      
      
        #math.equation(block: true, alt: "open parenthesis 1 , the fraction minus 5 π over 3 close parenthesis")[$( 1 , display(frac(− 5 π, 3)) )$]


      
      
        #math.equation(block: true, alt: "open parenthesis minus 3 , the fraction π over 2 close parenthesis")[$( − 3 , display(frac(π, 2)) )$]


        
          #figure(figph[polar plot], alt: "polar plot", caption: none)


        
      
      
        #math.equation(block: true, alt: "open parenthesis 2 , the fraction 7 π over 4 close parenthesis")[$( 2 , display(frac(7 π, 4)) )$]


      
      For Problems 5–8, convert the polar coordinates to Cartesian coordinates.


      
        #math.equation(block: true, alt: "open parenthesis 1 , the fraction 5 π over 4 close parenthesis")[$( 1 , display(frac(5 π, 4)) )$]


        
          #math.equation(block: true, alt: "open parenthesis the fraction minus the square root of 2 over 2 , the fraction minus the square root of 2 over 2 close parenthesis")[$( display(frac(− sqrt(2), 2)) , display(frac(− sqrt(2), 2)) )$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 0 , the fraction π over 12 close parenthesis")[$( 0 , display(frac(π, 12)) )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 3.4 , minus 1.5 close parenthesis")[$( 3.4 , − 1.5 )$]


        
          #math.equation(block: true, alt: "open parenthesis 0.241 , minus 3.391 close parenthesis")[$( 0.241 , − 3.391 )$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 5.6 , minus 1.1 close parenthesis")[$( − 5.6 , − 1.1 )$]


      
      For Problems 9–12, convert the Cartesian coordinates to polar coordinates with #math.equation(block: false, alt: "r greater than or equal to 0")[$r ≥ 0$] and #math.equation(block: false, alt: "0 less than or equal to θ less than or equal to 2 π")[$0 ≤ θ ≤ 2 π$].  Give exact values for #math.equation(block: false, alt: "r")[$r$] and #math.equation(block: false, alt: "θ")[$θ$].


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


        
          #math.equation(block: true, alt: "open parenthesis 3 the square root of 2 , the fraction 3 π over 4 close parenthesis")[$( 3 sqrt(2) , display(frac(3 π, 4)) )$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 0 , minus 2 close parenthesis")[$( 0 , − 2 )$]


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


        
          #math.equation(block: true, alt: "open parenthesis the square root of 29 , tan to the power minus 1 ⁡ open parenthesis the fraction minus 2 over 5 close parenthesis plus 2 π close parenthesis")[$( sqrt(29) , tan^(− 1) ( display(frac(− 2, 5)) ) + 2 π )$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 15 , minus 8 close parenthesis")[$( − 15 , − 8 )$]


      
      For Problems 13–16, sketch the region described by the inequalities.


      
        #math.equation(block: true, alt: "r greater than or equal to 0 , the fraction minus π over 4 less than or equal to θ less than or equal to the fraction π over 4")[$r ≥ 0 , " " display(frac(− π, 4)) ≤ θ ≤ display(frac(π, 4))$]


        
          #figure(figph[region on polar grid], alt: "region on polar grid", caption: none)


        
      
      
        #math.equation(block: true, alt: "1 less than or equal to r less than or equal to 3 , 0 less than or equal to θ less than or equal to π")[$1 ≤ r ≤ 3 , " " 0 ≤ θ ≤ π$]


      
      
        #math.equation(block: true, alt: "0 less than or equal to r less than or equal to 2")[$0 ≤ r ≤ 2$]


        
          #figure(figph[region on polar grid], alt: "region on polar grid", caption: none)


        
      
      
        #math.equation(block: true, alt: "r greater than or equal to 4")[$r ≥ 4$]


      
      For Problems 17–20, convert the equation into Cartesian coordinates.


      
        #math.equation(block: true, alt: "r equals 1")[$r = 1$]


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


        
      
      
        #math.equation(block: true, alt: "r equals minus 3 sec ⁡ open parenthesis θ close parenthesis")[$r = − 3 sec ( θ )$]


      
      
        #math.equation(block: true, alt: "r equals the fraction 6 over 1 minus 2 cos ⁡ open parenthesis θ close parenthesis")[$r = display(frac(6, 1 − 2 cos ( θ )))$]


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


        
      
      
        #math.equation(block: true, alt: "3 tan ⁡ open parenthesis θ close parenthesis equals 6 r sin ⁡ open parenthesis θ close parenthesis minus 1")[$3 tan ( θ ) = 6 r sin ( θ ) − 1$]


      
      For Problems 21–24, convert the equation into polar coordinates.


      
        #math.equation(block: true, alt: "x plus y equals 2")[$x + y = 2$]


        
          #math.equation(block: true, alt: "r cos ⁡ open parenthesis θ close parenthesis plus r sin ⁡ open parenthesis θ close parenthesis equals 2")[$r cos ( θ ) + r sin ( θ ) = 2$]


        
      
      
        #math.equation(block: true, alt: "the square root of x squared plus y squared equals 4 y")[$sqrt(x^(2) + y^(2)) = 4 y$]


      
      
        #math.equation(block: true, alt: "the fraction y over x equals the square root of x squared plus y squared")[$display(frac(y, x)) = sqrt(x^(2) + y^(2))$]


        
          #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis equals r")[$tan ( θ ) = r$]


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


      
      For Problems 25–28, use the catalog of polar graphs to help you identify and sketch the curve. Check your work by graphing with a calculator.


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


        
          Circle of radius 3 centered at the origin


        
      
      
        #math.equation(block: true, alt: "θ equals the fraction 3 π over 4")[$θ = display(frac(3 π, 4))$]


      
      
        #math.equation(block: true, alt: "r equals 6 cos ⁡ open parenthesis θ close parenthesis")[$r = 6 cos ( θ )$]


        
          Circle of radius 3 centered at #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]


        
      
      
        #math.equation(block: true, alt: "r squared equals 9 sin ⁡ open parenthesis 2 θ close parenthesis")[$r^(2) = 9 sin ( 2 θ )$]


      
      For Problems 29–32, write a polar equation for the graph.


      
        #figure(figph[polar grid], alt: "polar grid", caption: none)


        
          #math.equation(block: true, alt: "r equals 4")[$r = 4$]


        
      
      
        #figure(figph[polar grid], alt: "polar grid", caption: none)


      
      
        #figure(figph[polar grid], alt: "polar grid", caption: none)


        
          #math.equation(block: true, alt: "r equals 4 cos ⁡ open parenthesis θ close parenthesis")[$r = 4 cos ( θ )$]


        
      
      
        #figure(figph[polar grid], alt: "polar grid", caption: none)


      
      For Problems 33–36, find the coordinates of the intersection points of the two curves analytically. Then graph the curves to verify your answers.


      
        #math.equation(block: true, alt: "r equals 3 plus 2 sin ⁡ open parenthesis θ close parenthesis , r equals 4")[$r = 3 + 2 sin ( θ ) , " " r = 4$]


        
          #math.equation(block: true, alt: "open parenthesis 4 , the fraction π over 6 close parenthesis")[$( 4 , display(frac(π, 6)) )$], #math.equation(block: true, alt: "open parenthesis 4 , the fraction 5 π over 6 close parenthesis")[$" " ( 4 , display(frac(5 π, 6)) )$]


        
      
      
        #math.equation(block: true, alt: "r equals 3 cos ⁡ open parenthesis θ close parenthesis , r equals the square root of 3 sin ⁡ open parenthesis θ close parenthesis")[$r = 3 cos ( θ ) , " " r = sqrt(3) sin ( θ )$]


      
      
        #math.equation(block: true, alt: "r equals 4 sin ⁡ open parenthesis θ close parenthesis , r equals minus 4 cos ⁡ open parenthesis θ close parenthesis")[$r = 4 sin ( θ ) , " " r = − 4 cos ( θ )$]


        
          #math.equation(block: false, alt: "open parenthesis 2 the square root of 2 , the fraction 3 π over 4 close parenthesis")[$( 2 sqrt(2) , display(frac(3 π, 4)) )$] and the pole


        
      
      
        #math.equation(block: true, alt: "r equals 2 plus 6 sin ⁡ open parenthesis θ close parenthesis , r equals 4 sin ⁡ open parenthesis θ close parenthesis")[$r = 2 + 6 sin ( θ ) , " " r = 4 sin ( θ )$]


      
      For Problems 37–40, perform the indicated operations on the complex numbers.


      
        #math.equation(block: true, alt: "the fraction 5 minus 10 i over 2 minus i")[$display(frac(5 − 10 i, 2 − i))$]


        
          #math.equation(block: true, alt: "4 minus 3 i")[$4 − 3 i$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 4 minus 7 i close parenthesis open parenthesis 1 plus i close parenthesis")[$( 4 − 7 i ) ( 1 + i )$]


      
      
        #math.equation(block: true, alt: "5 i open parenthesis 2 minus i close parenthesis minus open parenthesis 7 plus 6 i close parenthesis")[$5 i ( 2 − i ) − ( 7 + 6 i )$]


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


        
      
      
        #math.equation(block: true, alt: "minus 8 plus 3 i plus the fraction 9 minus 4 i over i")[$− 8 + 3 i + display(frac(9 − 4 i, i))$]


      
      For Problems 41–44, evaluate the polynomial for the given values of the variable.


      
        #math.equation(block: true, alt: "z squared plus 4 z plus 6")[$z^(2) + 4 z + 6$]


        + #math.equation(block: false, alt: "z equals minus 2 plus i")[$z = − 2 + i$]
+ #math.equation(block: false, alt: "z equals minus 2 minus 1")[$z = − 2 − 1$]


        
          + #math.equation(block: false, alt: "1")[$1$]
+ #math.equation(block: false, alt: "1")[$1$]


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


        + #math.equation(block: false, alt: "z equals 3 minus 2 i")[$z = 3 − 2 i$]
+ #math.equation(block: false, alt: "z equals 3 plus 21")[$z = 3 + 21$]


      
      
        #math.equation(block: true, alt: "3 w squared minus 18 w plus 31")[$3 w^(2) − 18 w + 31$]


        + #math.equation(block: false, alt: "w equals 3 plus 4 i")[$w = 3 + 4 i$]
+ #math.equation(block: false, alt: "w equals 3 minus 4 i")[$w = 3 − 4 i$]


        
          + #math.equation(block: false, alt: "minus 44")[$− 44$]
+ #math.equation(block: false, alt: "minus 44")[$− 44$]


        
      
      
        #math.equation(block: true, alt: "2 w squared plus 8 w plus 11")[$2 w^(2) + 8 w + 11$]


        + #math.equation(block: false, alt: "w equals minus 2 minus 5 i")[$w = − 2 − 5 i$]
+ #math.equation(block: false, alt: "w equals minus 2 plus 51")[$w = − 2 + 51$]


      
      
        Verify that #math.equation(block: false, alt: "z sub 1 equals 2 plus i")[$z_(1) = 2 + i$] and #math.equation(block: false, alt: "z sub 2 equals 2 minus i")[$z_(2) = 2 − i$] are roots of the equation #math.equation(block: false, alt: "x squared minus 4 x plus 5 equals 0")[$x^(2) − 4 x + 5 = 0$].


        
          #math.equation(block: true, alt: "open parenthesis 2 plus or minus i close parenthesis squared minus 4 open parenthesis 2 plus or minus i close parenthesis plus 5 equals open parenthesis 4 plus or minus 4 i minus 1 close parenthesis minus open parenthesis 8 plus or minus 4 i close parenthesis plus 5 equals 0")[$( 2 ± i )^(2) − 4 ( 2 ± i ) + 5 = ( 4 ± 4 i − 1 ) − ( 8 ± 4 i ) + 5 = 0$]


        
      
      
        Verify that #math.equation(block: false, alt: "z sub 1 equals minus 3 plus 4 i")[$z_(1) = − 3 + 4 i$] and #math.equation(block: false, alt: "z sub 2 equals minus 3 minus 4 i")[$z_(2) = − 3 − 4 i$] are roots of the equation #math.equation(block: false, alt: "x squared plus 6 x plus 25 equals 0")[$x^(2) + 6 x + 25 = 0$].


      
      For Problems 47–50, expand the product of polynomials.


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


        
          #math.equation(block: true, alt: "z squared plus 4 z plus 5")[$z^(2) + 4 z + 5$]


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


      
      
        #math.equation(block: true, alt: "open bracket s plus open parenthesis 5 plus 4 i close parenthesis close bracket open bracket s plus open parenthesis 5 minus 4 i close parenthesis close bracket")[$[ s + ( 5 + 4 i ) ] [ s + ( 5 − 4 i ) ]$]


        
          #math.equation(block: true, alt: "s squared minus 10 s plus 41")[$s^(2) − 10 s + 41$]


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


      
      For Problems 51–52, sketch the set of points in the complex plane.


      
        #math.equation(block: true, alt: "z sub 1 equals minus 3 plus 2 i ,")[$z_(1) = − 3 + 2 i , " "$] #math.equation(block: true, alt: "z sub 2 equals minus 3 minus 2 i ,")[$z_(2) = − 3 − 2 i , " "$] #math.equation(block: true, alt: "z sub 3 equals 3 plus 2 i ,")[$z_(3) = 3 + 2 i , " "$] #math.equation(block: true, alt: "z sub 4 equals 3 minus 2 i")[$z_(4) = 3 − 2 i$]


        
          #figure(figph[complex numbers], alt: "complex numbers", caption: none)


        
      
      
        #math.equation(block: true, alt: "w sub 1 equals 4 minus 6 i ,")[$w_(1) = 4 − 6 i , " "$] #math.equation(block: true, alt: "w sub 2 equals 4 plus 6 i ,")[$w_(2) = 4 + 6 i , " "$] #math.equation(block: true, alt: "w sub 3 equals minus 4 minus 6 i ,")[$w_(3) = − 4 − 6 i , " "$] #math.equation(block: true, alt: "w sub 4 equals minus 4 plus 6 i")[$w_(4) = − 4 + 6 i$]


      
      For Problems 53–56,


      + Given one solution of a quadratic equation with rational coefficients, find the other solution.
+ Write a quadratic equation that has those solutions.


      
        #math.equation(block: true, alt: "minus 1 plus 7 i")[$− 1 + 7 i$]


        
          + #math.equation(block: false, alt: "minus 1 minus 7 i")[$− 1 − 7 i$]
+ #math.equation(block: false, alt: "x squared plus 2 x plus 50 equals 0")[$x^(2) + 2 x + 50 = 0$]


        
      
      
        #math.equation(block: true, alt: "2 minus 5 i")[$2 − 5 i$]


      
      
        #math.equation(block: true, alt: "3 minus the square root of 2 i")[$3 − sqrt(2) i$]


        
          + #math.equation(block: false, alt: "3 plus the square root of 2 i")[$3 + sqrt(2) i$]
+ #math.equation(block: false, alt: "x squared minus 6 x plus 11 equals 0")[$x^(2) − 6 x + 11 = 0$]


        
      
      
        #math.equation(block: true, alt: "4 plus the square root of 3 i")[$4 + sqrt(3) i$]


      
      For Problems 57–60, write the complex numbers in standard form. Give exact values for your answers.


      
        #math.equation(block: true, alt: "10 open parenthesis cos ⁡ open parenthesis the fraction minus π over 6 close parenthesis plus i sin ⁡ open parenthesis the fraction minus π over 6 close parenthesis close parenthesis")[$10 ( cos ( display(frac(− π, 6)) ) + i sin ( display(frac(− π, 6)) ) )$]


        
          #math.equation(block: true, alt: "5 the square root of 3 minus 5 i")[$5 sqrt(3) − 5 i$]


        
      
      
        #math.equation(block: true, alt: "8 open parenthesis cos ⁡ open parenthesis the fraction 5 π over 4 close parenthesis plus i sin ⁡ open parenthesis the fraction 5 π over 4 close parenthesis close parenthesis")[$8 ( cos ( display(frac(5 π, 4)) ) + i sin ( display(frac(5 π, 4)) ) )$]


      
      
        #math.equation(block: true, alt: "5 the square root of 2 open parenthesis cos ⁡ open parenthesis the fraction π over 4 close parenthesis plus i sin ⁡ open parenthesis the fraction π over 4 close parenthesis close parenthesis")[$5 sqrt(2) ( cos ( display(frac(π, 4)) ) + i sin ( display(frac(π, 4)) ) )$]


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


        
      
      
        #math.equation(block: true, alt: "6 the square root of 3 open parenthesis cos ⁡ open parenthesis the fraction minus π over 3 close parenthesis plus i sin ⁡ open parenthesis the fraction minus π over 3 close parenthesis close parenthesis")[$6 sqrt(3) ( cos ( display(frac(− π, 3)) ) + i sin ( display(frac(− π, 3)) ) )$]


      
      For Problems 61–66, write the complex numbers in polar form. Give exact values for your answers.


      
        #math.equation(block: true, alt: "3 minus 3 i")[$3 − 3 i$]


        
          #math.equation(block: true, alt: "3 the square root of 2 open parenthesis cos ⁡ open parenthesis the fraction 7 π over 4 close parenthesis plus i sin ⁡ open parenthesis the fraction 7 π over 4 close parenthesis close parenthesis")[$3 sqrt(2) ( cos ( display(frac(7 π, 4)) ) + i sin ( display(frac(7 π, 4)) ) )$]


        
      
      
        #math.equation(block: true, alt: "minus 4 minus 4 i")[$− 4 − 4 i$]


      
      
        #math.equation(block: true, alt: "minus 5")[$− 5$]


        
          #math.equation(block: true, alt: "5 open parenthesis cos ⁡ open parenthesis π close parenthesis plus i sin ⁡ open parenthesis π close parenthesis close parenthesis")[$5 ( cos ( π ) + i sin ( π ) )$]


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


      
      
        #math.equation(block: true, alt: "minus 1 minus the square root of 3 i")[$− 1 − sqrt(3) i$]


        
          #math.equation(block: true, alt: "2 open parenthesis cos ⁡ open parenthesis the fraction 4 π over 3 close parenthesis plus i sin ⁡ open parenthesis the fraction 4 π over 3 close parenthesis close parenthesis")[$2 ( cos ( display(frac(4 π, 3)) ) + i sin ( display(frac(4 π, 3)) ) )$]


        
      
      
        #math.equation(block: true, alt: "6 plus 2 the square root of 3 i")[$6 + 2 sqrt(3) i$]


      
      For Problems 67–70, find the product #math.equation(block: false, alt: "z sub 1 z sub 2")[$z_(1) z_(2)$] and the quotient #math.equation(block: false, alt: "the fraction z sub 1 over z sub 2")[$display(frac(z_(1), z_(2)))$].


      
        #math.equation(block: true, alt: "z sub 1 equals 8 open parenthesis cos ⁡ open parenthesis the fraction π over 6 close parenthesis plus i sin ⁡ open parenthesis the fraction π over 6 close parenthesis close parenthesis")[$z_(1) = 8 ( cos ( display(frac(π, 6)) ) + i sin ( display(frac(π, 6)) ) )$] #math.equation(block: true, alt: "z sub 2 equals 2 open parenthesis cos ⁡ open parenthesis the fraction 5 π over 6 close parenthesis plus i sin ⁡ open parenthesis the fraction 5 π over 6 close parenthesis close parenthesis")[$z_(2) = 2 ( cos ( display(frac(5 π, 6)) ) + i sin ( display(frac(5 π, 6)) ) )$]


        
          #math.equation(block: true, alt: "z sub 1 z sub 2, equals 16 open parenthesis cos ⁡ open parenthesis π close parenthesis plus i sin ⁡ open parenthesis π close parenthesis close parenthesis; equals minus 16 , the fraction z sub 1 over z sub 2; equals 4 open parenthesis cos ⁡ open parenthesis the fraction minus 2 π over 3 close parenthesis plus i sin ⁡ open parenthesis the fraction minus 2 π over 3 close parenthesis close parenthesis; equals minus 2 minus 2 the square root of 3 i")[$z_(1) z_(2) & = 16 ( cos ( π ) + i sin ( π ) ) \  & = − 16 , " " display(frac(z_(1), z_(2))) \  & = 4 ( cos ( display(frac(− 2 π, 3)) ) + i sin ( display(frac(− 2 π, 3)) ) ) \  & = − 2 − 2 sqrt(3) i$]


        
      
      
        #math.equation(block: true, alt: "z sub 1 equals 9 open parenthesis cos ⁡ open parenthesis the fraction minus 2 π over 3 close parenthesis plus i sin ⁡ open parenthesis the fraction minus 2 π over 3 close parenthesis close parenthesis")[$z_(1) = 9 ( cos ( display(frac(− 2 π, 3)) ) + i sin ( display(frac(− 2 π, 3)) ) )$]#math.equation(block: true, alt: "z sub 2 equals 3 open parenthesis cos ⁡ open parenthesis the fraction π over 3 close parenthesis plus i sin ⁡ open parenthesis the fraction π over 3 close parenthesis close parenthesis")[$z_(2) = 3 ( cos ( display(frac(π, 3)) ) + i sin ( display(frac(π, 3)) ) )$]


      
      
        #math.equation(block: true, alt: "z sub 1 equals 5 open parenthesis cos ⁡ open parenthesis the fraction minus 7 π over 12 close parenthesis plus i sin ⁡ open parenthesis the fraction minus 7 π over 12 close parenthesis close parenthesis")[$z_(1) = 5 ( cos ( display(frac(− 7 π, 12)) ) + i sin ( display(frac(− 7 π, 12)) ) )$]#math.equation(block: true, alt: "z sub 2 equals the fraction 1 over 2 open parenthesis cos ⁡ open parenthesis open parenthesis the fraction π over 4 close parenthesis plus i sin ⁡ open parenthesis the fraction π over 4 close parenthesis close parenthesis close parenthesis")[$z_(2) = display(frac(1, 2)) ( cos ( ( display(frac(π, 4)) ) + i sin ( display(frac(π, 4)) ) ) )$]


        
          #math.equation(block: true, alt: "z sub 1 z sub 2, equals the fraction 5 over 2 open parenthesis cos ⁡ open parenthesis the fraction minus π over 3 close parenthesis plus i sin ⁡ open parenthesis the fraction minus π over 3 close parenthesis close parenthesis; equals the fraction 5 over 4 minus the fraction 5 the square root of 3 over 2 i ,")[$z_(1) z_(2) & = display(frac(5, 2)) ( cos ( display(frac(− π, 3)) ) + i sin ( display(frac(− π, 3)) ) ) \  & = display(frac(5, 4)) − display(frac(5 sqrt(3), 2)) i ,$] #math.equation(block: true, alt: "the fraction z sub 1 over z sub 2, equals 10 open parenthesis cos ⁡ open parenthesis the fraction minus 5 π over 6 close parenthesis plus i sin ⁡ open parenthesis the fraction minus 5 π over 6 close parenthesis close parenthesis; equals minus 5 the square root of 3 minus 5 i")[$display(frac(z_(1), z_(2))) & = 10 ( cos ( display(frac(− 5 π, 6)) ) + i sin ( display(frac(− 5 π, 6)) ) ) \  & = − 5 sqrt(3) − 5 i$]


        
      
      
        #math.equation(block: true, alt: "z sub 1 equals 14 open parenthesis cos ⁡ open parenthesis the fraction 3 π over 2 close parenthesis plus i sin ⁡ open parenthesis the fraction 3 π over 2 close parenthesis close parenthesis")[$z_(1) = 14 ( cos ( display(frac(3 π, 2)) ) + i sin ( display(frac(3 π, 2)) ) )$]#math.equation(block: true, alt: "z sub 2 equals 2 open parenthesis cos ⁡ the fraction π over 6 plus i sin ⁡ the fraction π over 6 close parenthesis")[$z_(2) = 2 ( cos display(frac(π, 6)) + i sin display(frac(π, 6)) )$]


      
      For Problems 71–74, find the power.


      
        #math.equation(block: true, alt: "open parenthesis the fraction the square root of 3 over 2 plus the fraction 1 over 2 i close parenthesis to the power 12")[$attach(( display(frac(sqrt(3), 2)) + display(frac(1, 2)) i ), t: 12)$]


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


        
      
      
        #math.equation(block: true, alt: "open parenthesis 1 minus the square root of 3 i close parenthesis to the power 6")[$attach(( 1 − sqrt(3) i ), t: 6)$]


      
      
        #math.equation(block: true, alt: "open parenthesis minus the square root of 5 plus the square root of 5 i close parenthesis to the power minus 4")[$attach(( − sqrt(5) + sqrt(5) i ), t: − 4)$]


        
          #math.equation(block: true, alt: "the fraction minus 1 over 100")[$display(frac(− 1, 100))$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 1 minus i close parenthesis to the power minus 8")[$( − 1 − i )^(− 8)$]


      
      For Problems 75–78,


      + Find the roots and plot them in the complex plane.
+ Write the roots in standard form.


      
        The square roots of #math.equation(block: false, alt: "minus 16 i")[$− 16 i$]


        
          + #figure(figph[complex square roots], alt: "complex square roots", caption: none)
+ #math.equation(block: false, alt: "minus 2 the square root of 2 plus 2 the square root of 2 i")[$− 2 sqrt(2) + 2 sqrt(2) i$], #math.equation(block: false, alt: "2 the square root of 2 minus 2 the square root of 2 i")[$" " 2 sqrt(2) − 2 sqrt(2) i$]


        
      
      
        The cube roots of #math.equation(block: false, alt: "minus 8")[$− 8$]


      
      
        The cube roots of #math.equation(block: false, alt: "minus 27 i")[$− 27 i$]


        
          + #figure(figph[complex cube roots], alt: "complex cube roots", caption: none)
+ #math.equation(block: false, alt: "3 i")[$3 i$], #math.equation(block: false, alt: "the fraction minus 3 the square root of 3 over 2 minus the fraction 3 over 2 i")[$" " display(frac(− 3 sqrt(3), 2)) − display(frac(3, 2)) i$], #math.equation(block: false, alt: "the fraction 3 the square root of 3 over 2 minus the fraction 3 over 2 i")[$" " display(frac(3 sqrt(3), 2)) − display(frac(3, 2)) i$]


        
      
      
        The square roots of #math.equation(block: false, alt: "minus 2 plus 2 the square root of 3 i")[$− 2 + 2 sqrt(3) i$]


      
      For Problems 79–82, solve the equation.


      
        #math.equation(block: true, alt: "z to the power 6 plus 27 equals 0")[$z^(6) + 27 = 0$]


        
          #math.equation(block: false, alt: "3 open parenthesis cos ⁡ θ plus i sin ⁡ θ close parenthesis ,")[$3 ( cos θ + i sin θ ) ,$] for #math.equation(block: false, alt: "θ equals the fraction π over 6 ,")[$θ = display(frac(π, 6)) , " "$] #math.equation(block: false, alt: "the fraction π over 2 ,")[$display(frac(π, 2)) , " "$] #math.equation(block: false, alt: "the fraction 5 π over 6 ,")[$display(frac(5 π, 6)) , " "$] #math.equation(block: false, alt: "the fraction 7 π over 6 ,")[$display(frac(7 π, 6)) , " "$] #math.equation(block: false, alt: "the fraction 3 π over 2 ,")[$display(frac(3 π, 2)) , " "$] #math.equation(block: false, alt: "the fraction 11 π over 6")[$display(frac(11 π, 6))$]


        
      
      
        #math.equation(block: true, alt: "z to the power 4 minus 6 z squared plus 12 equals 0")[$z^(4) − 6 z^(2) + 12 = 0$]


      
      
        #math.equation(block: true, alt: "z to the power 4 plus 2 z squared plus 4 equals 0")[$z^(4) + 2 z^(2) + 4 = 0$]


        
          #math.equation(block: false, alt: "the square root of 2 open parenthesis cos ⁡ open parenthesis θ close parenthesis plus i sin ⁡ open parenthesis θ close parenthesis close parenthesis ,")[$sqrt(2) ( cos ( θ ) + i sin ( θ ) ) ,$] for #math.equation(block: false, alt: "θ equals the fraction π over 3 ,")[$θ = display(frac(π, 3)) , " "$] #math.equation(block: false, alt: "the fraction 2 π over 3 ,")[$display(frac(2 π, 3)) , " "$] #math.equation(block: false, alt: "the fraction 4 π over 3 ,")[$display(frac(4 π, 3)) , " "$] #math.equation(block: false, alt: "the fraction 5 π over 3")[$display(frac(5 π, 3))$]


        
      
      
        #math.equation(block: true, alt: "z to the power 6 minus 8 equals 0")[$z^(6) − 8 = 0$]