# Intermediate Algebra section 5.6 Factoring trinomials (quadratic forms)

Idea of trinomial factoring

(1) (x + b)(x + d) = x2+ (b+d) x + bd multiplying binomial by a binomial
(2) x2+ (b+d) x + bd = (x + b)(x + d) trinomial factoring
(3) (ax + b)(cx + d) = acx2+ (ad+bc) x + bd multiplying binomial by a binomial
(4) acx2+ (ad+bc) x + bd = (ax + b)(cx + d) trinomial factoring

Ex1, Factor :

(a) x2 - 3x + 2
(b) x2 -3x - 4
(c) x2 - 30x - 64
(d) x2 -19x+48
(e) x2 + 20x +36
(f) x2 +8x +64

Ex2, Factor

(a) 2x2+ 5x + 2
(b) 2x2 - 7x + 3
(c) 2x2 - 11x - 40
(d) 3x2 - 2x - 1
(e) 3x2 - 10x + 3
(f) 2x2 -13x + 6
(g) 3x2 +5x - 2

Ex3, Factor (completely)

(a) 2x2 - 4x - 96
(b) 3x2 +9x -54
(c) 4x2 +4x - 3
(d) 4x2 - x - 3
(e) 4x2 -12x +9
(f) 4x2 -12x - 40
(g) 4x2 -14x - 8
(h) 4x2 -17x + 4
(i) - x2 - 2x +8

Thm ( Test for factorability )
Quadratic form ax2 + bx + c is factorable if b2 - 4ac is a square of an integer (or rational number).

Ex4, Factor :

(a) 6x2 +x - 2
(b) 6x2 +3x - 2
(c) 6x2 +x - 2
(d) 6x2 - x - 35
(e) 6x2 - x - 22
(f) 9x2 +3x - 2
(g) 9x2 - 8x - 20
(h) 9x2 -18x +9
(i) 9x2 +10x - 40
(j) 9x2 + 82x +9
(k) 9x2 +30x +25

(a) x2 - 2x - 8
(b) x4 -2x2 - 8
(c) x6 -2x3 - 8
(d) 1 - 2x - 8x2
(e) x2 - 2xy - 8y2
(f) x2y2 -2xy - 8
(g) x4 -2x2y - 8y2
(h) x2 - 2xy2 - 8y4
(i) x6 - 2x3y2 - 8y4

Ex6, Factor :

(a) 12x2 - 4x - 16
(b) x3 - x2 - 20x
(c) 12x3 -10x2 - 8x
(d) 12x2y - 34xy2+14y3
(e) 3x3y +21x2y2 - 54xy3
(f) 12x2y -12xy2+3y3
(g) 12x2y -12xy2 - 24y3
(h) 2x2 - 8xy2 - 10y4
(i) x3 - 6x2 +9x

Ex7, ( factoring by grouping , either 2+2 or 3+1)

(a) 3xy - y - 6x + 2
(b) 2x2y - 3xy - 4x2 + 6x
(c) x2 - 4xy + 4 y2 - 9
(d) - x2 + 6x + 4 y2 - 9
(e) - 4x2 + 9y2 - 9 + 12x

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