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Factoring_Trinomials07
Intermediate Algebra
section 5.6 Factoring trinomials (quadratic forms)
Idea of trinomial factoring
(1) (x + b)(x + d) = x^{2}+ (b+d) x + bd multiplying binomial
by a binomial
(2) x^{2}+ (b+d) x + bd = (x + b)(x + d) trinomial factoring
(3) (ax + b)(cx + d) = acx^{2}+ (ad+bc) x + bd multiplying binomial by a binomial
(4) acx^{2}+ (ad+bc) x + bd = (ax + b)(cx + d) trinomial factoring
Ex1, Factor :
(a) x^{2}  3x + 2
(b) x^{2} 3x  4
(c) x^{2}  30x  64
(d) x^{2} 19x+48
(e) x^{2} + 20x +36
(f) x^{2} +8x +64
Ex^{2}, Factor
(a) 2x^{2}+ 5x + 2
(b) 2x^{2}  7x + 3
(c) 2x^{2}  11x  40
(d) 3x^{2}  2x  1
(e) 3x^{2}  10x + 3
(f) 2x^{2} 13x + 6
(g) 3x^{2} +5x  2
Ex3, Factor (completely)
(a) 2x^{2}  4x  96
(b) 3x^{2} +9x 54
(c) 4x^{2} +4x  3
(d) 4x^{2}  x  3
(e) 4x^{2} 12x +9
(f) 4x^{2} 12x  40
(g) 4x^{2} 14x  8
(h) 4x^{2} 17x + 4
(i)  x^{2}  2x +8
Thm ( Test for factorability )
Quadratic form ax^{2} + bx + c is factorable if b^{2}  4ac is a square of an integer
(or rational number).
Ex4, Factor :
(a) 6x^{2} +x  2
(b) 6x^{2} +3x  2
(c) 6x^{2} +x  2
(d) 6x^{2}  x  35
(e) 6x^{2}  x  22
(f) 9x^{2} +3x  2
(g) 9x^{2}  8x  20
(h) 9x^{2} 18x +9
(i) 9x^{2} +10x  40
(j) 9x^{2} + 82x +9
(k) 9x^{2} +30x +25
Ex5, Factoring different quadratic forms
(a) x^{2}  2x  8
(b) x^{4} 2x^{2}  8
(c) x^{6} 2x^{3}  8
(d) 1  2x  8x^{2}
(e) x^{2}  2xy  8y^{2}
(f) x^{2}y^{2} 2xy  8
(g) x^{4} 2x^{2}y  8y^{2}
(h) x^{2}  2xy^{2}  8y^{4}
(i) x^{6}  2x^{3}y^{2}  8y^{4}
Ex6, Factor :
(a) 12x^{2}  4x  16
(b) x^{3}  x^{2}  20x
(c) 12x^{3} 10x^{2}  8x
(d) 12x^{2}y  34xy^{2}+14y^{3}
(e) 3x^{3}y +21x^{2}y^{2}  54xy^{3}
(f) 12x^{2}y 12xy^{2}+3y^{3}
(g) 12x^{2}y 12xy^{2}  24y^{3}
(h) 2x^{2}  8xy^{2}  10y^{4}
(i) x^{3}  6x^{2} +9x
Ex7, ( factoring by grouping , either 2+2 or 3+1)
(a) 3xy  y  6x + 2
(b) 2x^{2}y  3xy  4x^{2} + 6x
(c) x^{2}  4xy + 4 y^{2}  9
(d)  x^{2} + 6x + 4 y^{2}  9
(e)  4x^{2} + 9y^{2}  9 + 12x
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