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CEGEP CHAMPLAIN - ST. LAWRENCE
201-NYA-05: Differential Calculus
Patrice Camiré
Factorization
1. Factor the following polynomials as much as possible.
(a) 3x2 − x
(f) −x2 + x + 12
(b) x2 − 2x − 15
(g) −x2 + 9
2
(c) 5x + 3x + 1
2
(k) x4 − 2x3 − 8x2
(l) x7 − 13x6 + 42x5
2
(h) −x + 5x + 24
(d) x + 6x + 5
(i) x2 − 18x + 81
(e) x2 − 6x + 8
(j) x2 + 9x + 20
(m) 3x2 − 6x − 24
(n) 3x2 − 4x − 4
2. Use long division to factor each polynomial as a product of two (or more) polynomials given a
specific zero.
(a) 2x2 − 3x − 2, zero: x = 2.
(c) x3 + 3x2 + 3x + 1, zero: x = −1.
(b) 3x2 + 17x + 20, zero: x = −4.
(d) −x4 + x3 + 3x2 − 3x − 4, zero: x = −1.
3. Factor the following polynomials as much as possible.
(a) x3 − 27
(c) x3 + 1
(b) x3 − 1
(d) x4 − 1
(e) x3 + 8
(f) x5 − 32
4. Factor and simplify the following expressions as much as possible.
(a) (2x + 1)3 (x − 4)2 + 3(2x + 1)2 (x − 4)3
(b) 2(3x − 2)8 (6x − 1)4 − 10(3x − 2)9 (6x − 1)3
(c) (4x + 3)1/2 (x + 5)11/4 − (4x + 3)5/2 (x + 5)3/4
(d) (x + 8)−1/5 (3x − 1)4/3 − (x + 8)9/5 (3x − 1)−2/3
5. Write each expression as a simplified fraction; factor the numerator and denominator as much as
possible.
1
2
−
(a) x + 12 x + 3
x −1
x
6
−
1 x+1
(b) x −
x2 + 3x − 10
(c)
(d)
x2 + 7x + 12
2
7
+
x+1 4−x
x2 − x − 30
x+2
x−1
−
x + 6 3x + 17
Answers
1. (a) x(3x − 1)
(h) −(x + 3)(x − 8)
(b) (x − 5)(x + 3)
(i) (x − 9)2
(c) 5x2 + 3x + 1
(j) (x + 4)(x + 5)
(d) (x + 1)(x + 5)
(k) x2 (x − 4)(x + 2)
(e) (x − 2)(x − 4)
(l) x5 (x − 6)(x − 7)
(f) −(x − 4)(x + 3)
(g) −(x − 3)(x + 3)
(m) 3(x − 4)(x + 2)
2
(n) 3(x − 2) x +
= (x − 2)(3x + 2)
3
2. (a) (x − 2)(2x + 1)
(b) (3x + 5)(x + 4)
(c) (x + 1)3
(d) (x + 1)(−x3 + 2x2 + x − 4)
3. (a) (x − 3)(x2 + 3x + 9)
(d) (x − 1)(x + 1)(x2 + 1)
(b) (x − 1)(x2 + x + 1)
(e) (x + 2)(x2 − 2x + 4)
(c) (x + 1)(x2 − x + 1)
(f) (x − 2)(x4 + 2x3 + 4x2 + 8x + 16)
4. (a) (5x − 11)(2x + 1)2 (x − 4)2
(b) −18(x − 1)(3x − 2)8 (6x − 1)3
2
8
x−
(4x + 3)1/2 (x + 5)3/4
(c) −15 x +
5
3
7
9
(d) 8 x +
x−
(x + 8)−1/5 (3x − 1)−2/3
4
2
5. (a)
(b)
−1
(x + 3)(x + 1)2
1
(c) − (x + 1)(x + 4)(x − 4)
5
x−3
(x − 1)(x + 1)(x + 5)
(d)
(x − 6)(x + 6)(3x + 17)
2(x + 4)