MTH 10905Algebra

Special Factoring Formulas and a

General Review of Factoring

Chapter 5 Section 5

Special Factoring

Describe the following.

a2 – b2

a3 + b3

a3 – b3

Special Factoring

Difference in Two Squares a2 – b2 = (a + b)(a – b)

Sum of Two Cubes a3 + b3 = (a + b)(a2 – ab + b2)

Difference of Two Cubes a3 – b3 = (a – b)(a2 + ab + b2)

Difference in Two Squares a2 – b2 = (a + b)(a – b)

Example: x2 – 100(x)2 – (10)2 a = x and b = 10(x + 10)(x – 10)

You can always check with the FOIL method

Example: 64x2 – 9(8x)2 – (3)2 a = 8x and b = 3 (8x + 3)(8x – 3)

Difference in Two Squares a2 – b2 = (a + b)(a – b)

Example: 49x2 – 81y2

(7x)2 – (9y)2 a = 7x and b = 9y (7x + 9y)(7x – 9y)

You can always check with the FOIL method

Example: 169x4 – 9y4

(13x2)2 – (3y2)2 a = 13x2 and b = 3y2

(13x2 + 3y2)(13x2 – 3y2)

Difference in Two Squares a2 – b2 = (a + b)(a – b)

Example: x6 – y8

(x3)2 – (y4)2 a = x3 and b = y4

(x3 + y4)(x3 – y4)

You can always check with the FOIL method

Example: 81x2 – 9y2 GCF = 99(9x2 – y2)

9[(3x)2 – (y)2] a = 3x and b = y 9(3x2 + y)(3x – y)

Difference in Two Squares a2 – b2 = (a + b)(a – b)

Example: p4 – 1(p2)2 – (1)2 a = p2 and b = 1(p2 + 1) (p2 – 1) (p2 + 1)(p + 1)(p – 1)

You can always check with the FOIL method

Also a difference in two squares(p)2 – (1)2

Sum of Two Cubes a3 + b3 = (a + b)(a2 - ab + b2)

Example: x3 + 125(x)3 + (5)3 a = x and b = 5(x + 5) (x2 – 5x + 52) (x + 5)(x2 – 5x + 25)

You can always check with the FOIL method

Example: 125x3 + 27y3 (5x)3 + (3y)3

a = 5x and b = 3y(5x + 3y) ((5x)2 – (3y)(5x) +

(3y)2) (5x + 3y)(25x2 – 15xy + 9y2)

Difference of Two Cubes a3 - b3 = (a - b)(a2 + ab + b2)

Example: z3 – 216 (z)3 – (6)3 a = z and b = 6(z – 6) (z2 + 6z + 62) (z – 6)(z2 + 6z + 36)

You can always check with the FOIL method

Example: 27m3 – n3

(3m)3 – (n)3

a = 3m and b = n(3m – n) [(3m)2 + (3m)(n) +

n2] (3m – n)(9m2 + 3mn + n2)

1. If all of the terms of the polynomial have a GCF other than 1, factor it out.

2. If the polynomial has two terms determine if it is a difference of two squares or a sum or difference of two cubes. Factor using formula.

3. If the polynomial has three terms, factor using any method we have discussed.

4. If the polynomial has more than three terms, try factoring by grouping.

5. Examine your factored polynomial to determine whether the terms in any factors have a common factor. If yes, factor out any common factors.

General Procedure for Factoring a Polynomial

Example: 5x4 – 405x2

GCF = 5x2

5x2(x2 – 81) 5x2(x + 9)(x – 9)

You can always check with the FOIL method

Example: 3p2 q2 + 12p2q – 36p2

GCF = 3p2

3p2(q2 + 4q – 12) (3p2)(q – 2)(q + 6)

General Procedure forFactoring a Polynomial

Difference in Two Squares

(x)2 – (9)2

Factor again

Factors -12 add 4

(-2)(6) -2 + 6

Example: 3t3r2 – 12tr2 + 15r2

GCF = 3r2

3r2(t3 – 4t + 5)

You can always check with the FOIL method

Example: 5rt + 5r – 15t -15

GCF = 55(rt + r – 3t – 3) 5 (r)(t + 1) – 3(t + 1)5(t + 1)(r – 3)

General Procedure forFactoring a Polynomial

PRIME

More than 3 terms so we can try Factor by Grouping at this point.

Factors 5 add -4

(1)(5) 1 + 5 = 6

(-1)(-5) -1 + -5 = -6

Example: 45x2 – 45x – 50GCF = 55(9x2 – 9x – 10)5(9x2 + 6x – 15x – 10)5 [(3x)(3x + 2) – (5)(3x + 2)]5(3x + 2)(3x – 5)

You can always check with the FOIL method

Example: 3m4n + 192mnGCF = 3mn3mn(m3 + 64)3mn(m3 + 43) 3mn(m + 4)(m2 – 4m +16)

General Procedure forFactoring a Polynomial

Factor by Grouping

a = 9 b = -9 c = -10

a • c = 9 • -10 = -90

Factor -90 add -9

(6)(-15) 6 + -15

Sum in Two Cubes

(a3 + b3)

(a + b)(a – ab +b2)

a = m b = 4

Remember

Learning to recognize special forms will help when you encounter them later.

There is no special form for the sum of two squares, only the difference of two squares has a special form.

When there are only two terms, you should try special forms for squares or cubes.

Try factor by grouping when you have more than three terms.

Sometimes trial and error is necessary in factoring, therefore the more you practice the more you will understand when to use trial and error and when to use another method.

HOMEWORK 5.5

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