3.7 Homework
1. Divide to show that * — 3 is a factor of each polynomial.
Confirm your answers using the factor theorem.
(a) *2 + * - 12
(b) *3 - 13* + 12
(c) -2x4 - 7*3 + 22*2 + 63* - 36
(d) *4 - 12*3 + 54*2 - 108* + 81
(e) 6*4 -M3*3 - 89*2 - 17* + 15
(f) xs + 2*4 - 18*3 - 36*2 + 81* + 162
12.
13.
2. State the remainder when * + 2 is divided into each polynomial,
(a) xi + 7* + 9
(b) 6*3 + 19*2 + 11* - 11
(c) *4 - 5*2 + 4
(d) *4 - 2*3 - II*2 + 10* - 2
(e) *3 + 3*2 - 10* +6
(f) 4*4 + 12*3 - 13*2 - 33* +
3. Determine whether 2* — 5 is a factor of each polynomial.
(a) 2*3 - 5*2 - 2* + 5
(b) 3*3 + 2*2 - 3* - 2
(c) 2*4 - 7*3 - 13*2 + 63* - 45 (d) 6*4 + *3 - 7*2 - * + 1
Factor fully,
(a) *3 - 64
(c) 64*3 + 1
(e) 8*9 + 216
(g)*6 + 1331
(b) y3 + 27
(d) 125*6 - 27
(f) 512-27*12
(h) 343*15 - 8
Factor by grouping,
(a) 2*3 + 2*2 + * + 1
(c) 8*3 + 12*2 + 2* + 3
(e) 6*5 - 2*4 - 9x2 + 3*
(b) 3*3 + 6*2 - * - 2
(d) ICk3 + 5*2 - 4* - 2
(f) 6*6 + 9*5 - 4* - 6
14. Graph/(*) = 2*3 - 3*2 - 3* + 2 using *-intercepts, the end behaviour of
/(*), and selected points on the graph.
LD
15. Verify your answer to question 14 using graphing technology.
16. Graph/(*) = -6*4 - 23*3 - 23*2 + 2* + 8 using *-intereepts, the end
behaviour of/(*), and selected points on the graph.
17. Verify your answer to question 16 using graphing technology.
4. Knowledge and Understanding: Which expression is a factor of
S*3 - 125: (3* + 2), (* - 5), or (2* - 5)? Justify your decision.
18. The polynomial 12*3 +fee2- * - 6 has factors 2* - 1 and 2* + 3.
5. Factor using the factor theorem.
(b) 4*3 + 12*2 - * - 15
(a) x3 - 3*2 - 10* + 24
(d) 4*4 + 7*3 - 80*2 - 21* + 270
(c) x4 + 8*3 + 4*2 - 48*
(e) x5 - 5x4 - Ix3 + 29*2 + 30* (f) x4 + 2*3 - 23*2 - 24* + 144
19. Application: When a*3 — *2 + 2* + b is divided by * — 1, the remainder i
20. The volume of a box is V(x) = *3 - 15*2 + 66* - 80.
6. Factor fully.
(a) *3 + 9*2 + 8* - 60
(c) *4 - 5*2 + 4
(e) x3 - x2 + * -1
21. Thinking. Inquiry, Problem Solving: Determine a general rule to help decide
whether (* - a) and (* + a) are factors of*" — a" and x" + a".
Determine the value of k.
(b) *3 - 7* - 6
(d) *4 + 3*3 - 38*2 + 24* + 64
(f) *5 -
4
2*3 - 2*2
7. Communication: Suppose that/(*) is a cubic polynomial function with
integral coefficients. Describe how you could find a zero and use it to find
any other zeros that/(*) might have.
gg 23. Verify your answer to question 22 using graphing technology.
A 24.
9. Factor using the factor theorem.
(a) 6*3 + 5*2 - 21* + 10
(b) 9*3 - 3*2 - 41* + 35
(c) 6*4 - 19*3 - 2*2 + 44* - 24 (d) 10*4 + 13*3 - 43*2 - 52* + 12
(e) 8*3 + 12*2 - 2* - 3
(f) 30*3 - *2 - 6* + 1
11. (a) Use the factor theorem to factor.
iv. 8*3 + 125
i. *3 + 1
ii. *3 + 8
Hi r3 + 64
(b) A polynomial in the form «3 + b3 is called a sum of cubes.
Use the pattern of factoring in (a) to factor a3 + fo3.
(c) Use the result of (b) to factor 125*3 + 64. Check by multiplying the
factors.
(a) Determine expressions for the dimensions of the box in terms of *.
(b) Graph the volume function and indicate any restrictions on *.
(c) Explain why * = 7 is inadmissible in the context of the question.
22. Check Your Understanding: Determine the factors of
/(*) = 2*4 - *3 - 14*2 - 5* + 6.
8. Determine whether or not the given value is a zero of/(*). If the value is a
zero, determine any other zeros the function might have.
(a) /(*) = 2*3 + *2 - 13* + 6, * = 0.5
(b) /(*) - 6*3 + 17*2 - 4* - 3, * = -2.25
(c) /(*) = *3 - 2*2 - 21* - 18, * = 6
(d) /(*) = Zt4 - *3 - 26*2 - 11* + 12, * = 4
(e) /(*) = *4 - 3*3 + 3*2 - 3* + 2, * = 2
(f) /(*) = 3*4 - 2*3 + 5*2 - 2* + 7, * = -3
10. (a) Use the factor theorem to factor.
i. *3 - 1
ii. *3 - 27
Hi. *3 - 125
iv. 8*3 - 27
(b) A polynomial in the form a3 - b3 is called a difference of cubes.
Use the pattern of factoring in (a) to factor a3 — b3.
(c) Use the result of (b) to factor 64*3 - 27. Check by multiplying the
factors.
10. When it is divided by * — 2, the remainder is 51. Find a and b.
The graph of/(*) = ax4 + far2 + c* - 24 crosses the *-axis at 1, -2,
and 3.
(a) Determine all zeros of/(*).
(b) Graph/(*) and comment on its end behaviour.
(c) State the coordinates of all turning points and indicate whether each
turning point is a local maximum or local minimum for/(*).
~
3.7 Homework Solutions
<»>-$
«>-34
(a) yes
ft) -5
(e) X
<h)no
(2* - 5}
(») {.r 4- 3
<£>.
- 2X.v - 4)
4- 6X-f - 2X* -1- 4)
(a> (x 4- 6X* -r 5X* - 2)
(c)0
12. (a) (r - 4XJT2 •*- 4x + 16)
(c)
(» Ox r 3X2* + SXx - i)
(d) (x 4- 5X* +• 2X*( - 9X* - 3
(b) (y 4- 3X>'2 - ^y 4 9)
(c) (4x + IXlfe 2 - 4or 4- I) (d) (Sx2 - 3)(25x4 4 15x- 4- 9)
(e) 8Cx3 + 3)(x6 - 3.V3 + 9)
(£) (8 - 3.*4)(64 4 24i^ 4- 9xs)
(g) (x24- IlXx 4 - Hx 2 4 121)
(h) 0*3 - 2X49*'° + I4x! 4- 4)
13. (a) (A 4 U(2x2 + I)
(b) (x 4 2)(3x2 - 1)
(C) (It 4 3X4*2 4- 1)
(d) C2x 4- IXSx2 - 2)
(B) x(3x - 1X2*3 - 3)
(I) (2x 4 SJtSx5 - 2)
W»> Or * 2'Kx 4- IX* - 3)
(d) (x + S)(x + I)(x - 2)(x - 4)
(e) (x - l)(x2 4- 1)
(f) (x - l)(x5 4 I)2
Use the factors of the constant term, the leading coefficient and the
Rational Zero Test and evaluate using the Factor Theorem,
(a) yes: other zeros: —3 and 2
(b) no
(c) yes; other zeros: — 1 and —3
(d) yes: other zeros: —3, —1 and 0.5
(e) yes; other zero: I
(a)
(c)
(d)
(e)
10. (a)
(b)
(C)
11. (a)
(b)
(2x 4- 5)(3x - 2)(x - 1)
(x - 2)2(3.f - 2X2* 4 3)
(5* - IXx - 2)(2* + 3)(x
(2* 4- 3X2* 4- l)(2x - 1)
(i) (x - l)(x2 4- x 4 l)
(ffi) (x - 5X.f2 + 5x 4- 25)
(a - h)(a2 4 ab + Ir)
<4x - 3)(lfo2 4- 12* 4- 9)
(i)(x 4 1X*: -x + 1)
(iU) (x 4- 4)(.v2 - 4x 4 16)
(a + b)(a- - ab 4- Ir)
(f) no
(b) Ox + 7)(.v - l)(3x - 5)
4- 2)
(f) (2* 4 lX3x - l)(5x - 1)
(U) (x - 3)(x2 4- 3x + 9)
(iv) (2* - 3X4x2 4 6x + 9)
(ii) (x + 2)(Jt2 - 2x 4- 4)
(iv) (2x 4- 5X4x2 - IQx +25)
(c) (5* + 4)(25xa - 20* 4 16)
i = 6. fc = 3
20. (a) (x - 2), (x - 5), (x - S)
(b)
„
x >!
(c) x = 7 is not in the domain
21. For x" — a", (x — a) is a factor. If n is even, then (x 4 a) is also a
factor. For x" + a", (x — a) is not a factor. If n is odd. then
(x 4 a) is a factor.
22. /(x) = (x 4 2Xx + lX2x - l)(x - 3)
24. (a) -2,1 and 3
(b) x > -»», y -» -«•>; jt -» 4- •>.
turning points ami local maximums:
(-2,0) and (2.225, 33.894)
turning point and local minimum:
(-0.225, -24.894)