3.3 special products and factoring

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


222 5ba

29ba

4224 2510 bbaa

81182 22 bababa

8118182 22 bababa

222222 552 bbaa

22 992 baba


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


22 25309 yxyx

bbb 462 23

253 yx

222 2 bbb

22 5303 yxyx


3324 567 yxyx

22425 23 xxx

yxyx 87 23

124252 2 xxx


33 23 xxx

abxbax 2332 22

332 xxx

bbxaax 3322 22

33 23 xxx

13 2 xx

1312 22 xbxa )32(12 bax


60273 24 xx

33 6427 yxyx

4153 22 xx

3333 43 yxyx

3333 43 yxyx


14 a

11 22 aa 111 2 aaa

More EXERCISES for YOU!


zxyzxy 2323

11 baba

1414 yxyx

26a

222 5ba


232 ba yx

29ba

352 x

325 94 yx


yzxzyxzyx 22334 363

22 3512 yxyx

22 25309 yxyx

bbb 462 23

3324 567 yxyx


423324 31218 yxyxyx

22425 23 xxx

33 23 xxx

2 22 3 3 2ax b bx a

21772 mm


3223 2615 xyyxyx

60273 24 xx

121444 2 aa xx

33 6427 yxyx

14 a

END OF 3.3

3.3 special products and factoring

Documents

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