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Math 31
Answers to Quiz 7
(Factoring polynomials:)
(1) x4 − 16 = (x2 )2 − (4)2 = (x2 + 4)(x2 − 4) = (x2 + 4)(x + 2)(x − 2)
(2) 5x2 − 20x + 20 = 5(x2 − 4x + 4) = 5(x − 2)2
(3) x3 + 27 = (x)3 + (3)3 = (x + 3)(x2 − 3x + 9)
(4) 2a2 + a − 6 = (2a − 3)(a + 2)
(5) x2 − 18x + 81 = (x − 9)2
(6) a4 b4 − c4 = (a2 b2 + c2 )(a2 b2 − c2 ) = (a2 b2 + c2 )(ab + c)(ab − c)
(7) 5x4 − 25x3 + 30x2 = 5x2 (x2 − 5x + 6) = 5x2 (x − 2)(x − 3)
(8) xy − 2x + 3y − 6 = (xy − 2x) + (3y − 6) = x(y − 2) + 3(y − 2) = (y − 2)(x + 3)
(9) a9 −a = a(a8 −1) = a(a4 +1)(a4 −1) = a(a4 +1)(a2 +1)(a2 −1) = a(a4 +1)(a2 +1)(a+1)(a−1)
(10) Solve the equation 2x2 + 5x − 3 = 0 using Factoring.
Factor the left hand side:
=⇒ (2x − 1)(x + 3) = 0
Either 2x − 1 = 0 =⇒ x =
The solution set is { 12 ,
(11) (A) x − 3;
(12) (D) 7x − 4;
(13) (B) x + 4;
1
2
Or,
x + 3 = 0 =⇒ x = −3
−3 }
x2 − x − 6 = (x − 3)(x + 2)
492 − 16 = (7x)2 − 42 = (7x + 4)(7x − 4)
3x2 + 5x − 28 = (3x − 7)(x + 4)
(14) (B) 3x + 4;
27x3 + 64 = (3x)3 + (4)3 = (3x + 4)((3x)x − (3x)(4) + 42 ) = (3x + 4)(9x2 − 12x + 16)
(15) (C) x + 3;
x4 − 81 = (x2 )2 − 92 = (x2 + 9)(x2 − 9) = (x2 + 9)(x + 3)(x − 3)
(16) (A) x − 2;
x3 − 3x2 − 4x + 12 = (x3 − 3x2 ) − (4x − 12) = x2 (x − 3) − 4(x − 3)
= (x − 3)(x2 − 4) = (x − 3)(x + 2)(x − 2)
(17) (C) x2 + 3x + 9;
x3 − 27 = x3 − 33 = (x − 3)(x2 + 3x + 9)
(18) (D) { 3 , −9 }:
Solve the equation x2 + 6x = 27.
x2 + 6x = 27 =⇒ x2 + 6x − 27 = 0 =⇒ (x + 9)(x − 3) = 0 =⇒ x = −9, x = 3
(19) (D) { −2 , 12 }:
Solve the equation x(x − 10) = 24.
x(x − 10) = 24 =⇒ x2 − 10x − 24 = 0 =⇒ (x − 12)(x + 2) = 0 =⇒ x = 12, x = −2
(20) (C) { 3 ,
6
7
}:
Solve the equation 7x2 − 27x + 18 = 0.
7x2 − 27x + 18 = 0 =⇒ (7x − 6)(x − 3) = 0 =⇒ x = 67 , x = 3
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