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

Ex5, Factoring different quadratic forms

(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