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3.3
SPECIAL PRODUCTS
AND FACTORING
Special Products and Factoring
Upon completion, you should be able to
Find special products
Factor a polynomial completely
Special Products
rules for finding products in a faster way
long multiplication is the last resort
Product of two binomials
Examples:
1) (x + 5)(x + 6)
2) (2x 3)(x 7)
3) (3x 5)(2x + 3)
2ax b cx d acx ad bc x bd
2 11 30x x
22 17 21x x26 15x x
Square of a binomial
Examples:
1) (x + 2y)2
2) (2x 3y)2
3) (3w2 4v3)2
2 2 22 a b a ab b
2 24 4x xy y 2 24 12 9x xy y 4 2 3 69 24 16w w v v
2 2 22 a b a ab b
Product of sum and difference
Examples:
1) (3x + 5y)(3x 5y)
2) (2x+3y2)(2x 3y2)
2 2a b a b a b
2 29 25x y 2 44 9x y
Cube of a binomial
Examples:
1) (x + y2)3
2) (2x 3y)3
3 3 2 2 33 3 a b a a b ab b
3 3 2 2 33 3 a b a a b ab b
3 2 2 4 6= x 3 3x y xy y
2 2 33= 8x 3 2 3 3 2 3 3x y x y y
3 2 2 3= 8x 36 54 27x y xy y
Product of a binomial and a trinomial
Examples:
1) (x + y2)(x2 xy2 + y4)
2) (2x 3y)(4x2 + 6xy + 9y2)
2 2 3 3 a b a ab b a b
2 2 3 3 a b a ab b a b
3 6x y
3 38 27x y
Square of a trinomial
Examples:
1) (x + y2 + w)2
2) (2x 3y + z)2
2 2 2 2 2 2 2a b c a b c ab ac bc
2x 4y 2w 22xy 2xw22y w
24x 29y 12xy2z
4xz 6yz
FACTORING
process of writing an expression as product of its factors
reverse process of finding products
Factoring
A polynomial with integer coefficients is said to be prime if it has no monomial or
polynomial factors with integer coefficients
other than itself and one.
A polynomial with integer coefficients is said to be factored completely when
each of its polynomial factor is prime.
Factoring a common monomial factor
Examples:
1) 8x2 + 4x
2) x5 3x4 + x3
ab ac a b c
4 x 1
3x 3x 1
2x
2x
Difference of two squares
Examples:
1) 16 y2
2) 4u2 9v2
3) (a+2b)2 (3b + c)2
2 2a b a b a b
2 24 y 4 y 4 y
2 2
2 3u v 2u 3v 2u 3v
5a b c a b c 2a b 3b c 2 3a b b c
Factoring trinomials
Examples:
1) x2 + 7x + 10
2) a2 10a + 24
3) a2 + 4ab 21b2
2 ( )acx ad bc x bd ax b cx d
x x 5 2 a a 6 4
a a b b7 3
Perfect square trinomials
Examples:
1) y2 10y + 25
2) 16x2 + 8x + 1
3) 9x2 30xy + 25y2
22 2
22 2
2
2
a ab b a b
a ab b a b
2
5y
2
4 1x
2
3 5x y
Sum and difference of cubes
Examples:
1) 27 x3
2) t3 + 8
3) y6 2y3 + 1
4) x6 2x3y3 + y6
3 3 2 2
3 3 2 2
a b a b a ab b
a b a b a ab b
3 33 x 3 x 9 3x 2x3 32t t 2 2t 2t 4
Factoring by GROUPING
Examples:
1) 10a3 + 25a 4a2 10
3 2
2 2
2
= 10a 25 4 10
5 2 5 2 2 5
2 5 5 2
a a
a a a
a a
Factoring by GROUPING
Examples:
1) 3xy yz + 3xw zw
Special Products and Factoring
Summary
Always look for a common monomial factor, FIRST.
Factoring can also be done by grouping some terms to yield a common polynomial
factor.
SOLVED EXERCISES
Find the following products.
zxyzxy 2323
1414 yxyx
222 49 zyx
22 14 yx
1816 22 yyx
22 23 zxy
1816 22 yyx
Find the following products.
222 5ba
29ba
4224 2510 bbaa
81182 22 bababa
8118182 22 bababa
222222 552 bbaa
22 992 baba
Find the following products.
352 x
2223 55235232 xxx
125150608 23 xxx
352 x
2223 55235232 xxx
125150608 23 xxx
Factor the following completely.
yzxzyxzyx 22334 363
22 3512 yxyx
22 13 xyyzx
yxyx 57
123 222 xyyxyzx
Factor the following completely.
22 25309 yxyx
bbb 462 23
253 yx
222 2 bbb
22 5303 yxyx
Factor the following completely.
3324 567 yxyx
22425 23 xxx
yxyx 87 23
124252 2 xxx
Factor the following completely.
33 23 xxx
abxbax 2332 22
332 xxx
bbxaax 3322 22
33 23 xxx
13 2 xx
1312 22 xbxa )32(12 bax
Factor the following completely.
60273 24 xx
33 6427 yxyx
4153 22 xx
3333 43 yxyx
3333 43 yxyx
Factor the following completely.
14 a
11 22 aa 111 2 aaa
More EXERCISES for YOU!
Find the following products.
zxyzxy 2323
11 baba
1414 yxyx
26a
222 5ba
Find the following products.
232 ba yx
29ba
352 x
325 94 yx
Factor the following completely.
yzxzyxzyx 22334 363
22 3512 yxyx
22 25309 yxyx
bbb 462 23
3324 567 yxyx
Factor the following completely.
423324 31218 yxyxyx
22425 23 xxx
33 23 xxx
2 22 3 3 2ax b bx a
21772 mm
Factor the following completely.
3223 2615 xyyxyx
60273 24 xx
121444 2 aa xx
33 6427 yxyx
14 a
Factor the following completely.
3223 2615 xyyxyx
60273 24 xx
121444 2 aa xx
33 6427 yxyx
14 a
END OF 3.3