THE FACTOR THEOREM
1. Determine if x – 2 is a factor.
a) 3x3 – x2 + 2x b) x3 + 2x2 – 16 c) 2x3 – 3x2 + x – 6
2. List the values that could be zeros of each polynomial. Then, factor the polynomial.
a) x3 – 2x2 – x + 2 b) x3 – 7x – 6 c) x3 + 5x2 – 2x – 24
3. Factor each polynomial by grouping terms.
a) x3 + 2x2 – 9x – 18 b) 2x3 + 5x2 – 8x – 20
c) x3 – 3x2 – 25x + 75 d) 3x3 – 5x2 – 27x + 45
4. Determine the values that could be zeros of each polynomial. Then, factor the polynomial.
a) x3 + x2 – 10x + 8 b) 2x3 + 5x2 + x – 2
c) 2x3 + 3x2 – 5x – 6 d) 3x3 – 16x2 + 23x – 6
5. Factor each polynomial.
a) x3 + 5x2 – x – 5 b) x3 – 7x + 6 c) x3 – 3x2 – 4x + 12
d) x4 + 4x3 – x2 – 16x – 12 e) x4 – 3x3 – 14x2 + 48x – 32
6. Determine the value of k so that x – 3 is a factor of x3 – 2x2 + kx – 6.
7. Determine the value of k so that 2x + 5 is a factor of 4x3 – kx2 – 6x + 10.
8. A carpenter is building a rectangular storage shed whose volume, V, in cubic metres, can be
modelled by V(x) = 4x3 – 36x2 + 107x – 105.
a) Determine the possible dimensions of the shed, in terms of x, in metres, that result in the
volume in part a).
b) What are the dimensions of the shed when x = 5.2?
9. Factor each polynomial.
a) 2x3 + 11x2 + 2x – 15 b) 3x3 + 8x2 + 3x – 2
c) 5x3 – 17x2 + 16x – 4 d) 4x3 + 5x2 – 23x – 6
10. Factor each polynomial.
8
a) 8x3 – 125 b) 64x3 + c) 216x3 + y3
27
1
d) 27 – t6 e) 125x6 – y3 f) 8x6 + 343y12
64
11. Factor each polynomial by letting t = x2.
a) 16x4 – 17x2 + 1 b) 9x4 – 61x2 + 100
12. Factor.
3x5 – 2x4 – 22x3 – 4x2 + 19x + 6
Answers
1. a) not a factor b) factor c) factor 2. a) 1, 2; (x – 2)(x – 1)(x + 1) b) 1, 2, 3, 6; (x + 2)(x + 1)(x – 3)
c) 1, 2, 3, 4, 6, 8, 12, 24; (x + 4)(x + 3)(x – 2) 3. a) (x + 2)(x – 3)(x + 3) b) (2x + 5)(x – 2)(x + 2)
c) (x – 3)(x – 5)(x + 5) d) (3x – 5) (x – 3)(x + 3) 4. a) 1, 2, 4, 8; (x – 1)(x – 2)(x + 4)
b) 1, 2, ; (2x –1)(x + 1)(x + 2) c) 1, 2, 3, 6, ± ,± ; (x + 1)(x + 2)(2x – 3)
1 1 3
2 2 2
d) 1, 2, 3, 6, ± , ± ; (3x – 1)(x – 2)(x – 3) 5. a) (x + 1)(x – 1)(x + 5) b) (x – 1)(x – 2)(x + 3)
1 2
3 3
c) (x – 3)(x + 2)(x – 2) d) (x + 1)(x + 2)(x – 2)(x + 3) e) (x – 1)(x – 2)(x + 4)(x – 4)
6. –1 7. –6 8. a) (x – 3)(2x – 5)(2x – 7) b) 2.2 m by 5.4 m by 3.4 m
9. a) (2x + 3)(x – 1)(x + 5) b) (3x – 1)(x + 1)(x + 2) c) (5x – 2)(x – 1)(x – 2) d) (4x + 1)(x + 3)(x – 2)
2 8 4
4 x + 16 x − x + c) (6x + y)(36x2 – 6xy + y2) d) (3 – t2)(9 + 3t2 + t4)
2
10. a) (2x – 5)(4x2 + 10x + 25)b)
3 3 9
2 1 5 2 1 2
e) 5 x − y 25 x + x y + y f) (2x2 + 7y4)(4x4 –14x2y4 + 49y8)
4
4 4 16
11. a) (4x – 1)(4x +1)(x – 1)(x + 1) b) (3x – 5)(3x + 5)(x – 2)(x + 2) 12. (x – 1)(x +2)(x – 3)(3x + 1)(x + 1)
