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For Common Assessment Adding and Subtracting Polynomials Multiplying Polynomials Factoring Polynomials

# For Common Assessment Adding and Subtracting Polynomials Multiplying Polynomials Factoring Polynomials

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For Common Assessment

Multiplying Polynomials

Factoring Polynomials

1) Align The Like Terms

2) Add/Subtract The Like Terms*Subtracting is the same as adding the opposite!!

** When adding or subtracting, EXPONENTS STAY THE SAME!!

There are two ways to add and subtract polynomials. You can do it horizontally or vertically.

•Simplify (2z + 5y) + (3z – 2y)

(2z + 5y) + (3z – 2y)

=  2z + 5y + 3z – 2y

=  2z + 3z + 5y – 2y

=  5z + 3y

Horizontal example:

Line up your like terms. 9y – 7x + 15a

+ -3y + 8x – 8a_________________________

Add the following polynomials (9y – 7x + 15a) + (-3y + 8x – 8a)

6y + x + 7a

3a2 + 3ab – b2

+ 4ab + 6b2

_________________________

Add the following polynomials (3a2 + 3ab – b2) + (4ab + 6b2)

3a2 + 7ab + 5b2

Line up your like terms. 4x2 – 2xy + 3y2

+ -3x2 – xy + 2y2

_________________________

x2 - 3xy + 5y2

Add the following polynomials (4x2 – 2xy + 3y2) + (-3x2 – xy + 2y2)

9y – 7x + 15a+ (+ 3y – 8x + 8a)--------------------------------------

Subtract the following polynomials (9y – 7x + 15a) – (-3y +8x – 8a)

12y – 15x + 23a

7a – 10b+ (– 3a – 4b)--------------------------------------

Subtract the following polynomials (7a – 10b) – (3a + 4b)

4a – 14b

4x2 – 2xy + 3y2

+ (+ 3x2 + xy – 2y2)--------------------------------------

7x2 – xy + y2

Subtract the following polynomials (4x2 – 2xy + 3y2) – (-3x2 – xy

+ 2y2)

Subtract (5x2 + 3a2 – 5x) – (2x2 – 5a2 + 7x)

5x2 + 3a2 – 5x

+ (- 2x2 + 5a2 – 7x)

--------------------------------------

3x2 + 8a2 – 12x

Subtract (3x2 + 8x + 4) – (5x2 – 4)

3x2 + 8x + 4

+ (- 5x2 + 4)

--------------------------------------

-2x2 + 8x + 8

Find the sum or difference.(5a – 3b) + (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 3b

Find the sum or difference.(5a – 3b) – (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 9b

(5x2 - 3x + 7) + (2x2 + 5x - 7)

= 7x2 + 2x

(3x3 + 6x - 8) + (4x2 + 2x - 5)

= 3x3 + 4x2 + 8x - 13

(2x3 + 4x2 - 6) – (3x3 + 2x - 2)

(7x3 - 3x + 1) – (x3 - 4x2 - 2)

(2x3 + 4x2 - 6) + (-3x3 + -2x - -2)

= -x3 + 4x2 - 2x - 4

(7x3 - 3x + 1) + (-x3 - -4x2 - -2)

= 6x3 + 4x2 - 3x + 3

7y2 – 3y + 4 + 8y2 + 3y – 4

2x3 – 5x2 + 3x – 1 – (8x3 – 8x2 + 4x + 3)

–6x3 + 3x2 – x – 4

= 15y2

(7y3 +2y2 + 5y – 1) + (5y3 + 7y)

12y3 + 2y2 + 12y – 1

(b4 – 6 + 5b + 1) + (8b4 + 2b – 3b2)

= 9b4 – 3b2 + 7b – 5

Remember that when you multiply two powers with the same bases, you add the exponents.

(5m2n3)(6m3n6)

5 · 6 · m2+3n3+6

30m5n9

Pre-Algebra

To multiply two monomials, multiply the coefficients and add the exponents of the variables that are the same.

MULTIPLYING POLYNOMIALS

Multiply.

Multiplying Monomials

A. (2x3y2)(6x5y3)

(2x3y2)(6x5y3)

12x8y5

B. (9a5b7)(–2a4b3)

(9a5b7)(–2a4b3)

–18a9b10

Pre-Algebra

Try This

Multiply.

A. (5r4s3)(3r3s2)

(5r4s3)(3r3s2)

15r7s5

B. (7x3y5)(–3x3y2)

(7x3y5)(–3x3y2)

–21x6y7

Multiply.

Multiplying a Polynomial by a Monomial

A. 3m(5m2 + 2m)

3m(5m2 + 2m)

15m3 + 6m2

Multiply each term in parentheses by 3m.

B. –6x2y3(5xy4 + 3x4)

–6x2y3(5xy4 + 3x4)

–30x3y7 – 18x6y3

Multiply each term in parentheses by –6x2y3.

Multiply.

Multiplying a Polynomial by a Monomial

C. –5y3(y2 + 6y – 8)

–5y3(y2 + 6y – 8)

–5y5 – 30y4 + 40y3

Multiply each term in parentheses by –5y3.

Pre-Algebra

Try This: Example 2A & 2B

Multiply.

Insert Lesson Title Here

A. 4r(8r3 + 16r)

4r(8r3 + 16r)

32r4 + 64r2

Multiply each term in parentheses by 4r.

B. –3a3b2(4ab3 + 4a2)

–3a3b2(4ab3 + 4a2)

–12a4b5 – 12a5b2

Multiply each term in parentheses by –3a3b2.

Example 2

Insert Lesson Title Here

Multiply.

C. –2x4(x3 + 4x + 3)

–2x4(x3 + 4x + 3)

–2x7 – 8x5 – 6x4

Multiply each term in parentheses by –2x4.

Pre-Algebra

Multiply. (2x + 3)(5x + 8)

Using the Distributive property, multiply

2x(5x + 8) + 3(5x + 8).

10x2 + 16x + 15x + 24

Combine like terms.

10x2 + 31x + 24

Another option is called the FOIL method.

F.O.I.L

F irst

O uter

I nner

L ast

( x + 2 ) ( x + 5 )

EXAMPLES

( x + 4 ) ( x + 8 ) =

( x + 5 ) ( x – 6) =

x2 + 8x + 4x + 32

x2 − 6x + 5x − 30

x2 + 12x + 32

x2 + 12x + 32

x2 − x − 30 x2 − x − 30

PRACTICE

( x − 7 ) ( x − 4 ) =

( x + 10 ) ( x + 3 ) =

x2 + 3x + 10x + 30

x2 + 13x + 30

x2 + 3x + 10x + 30

x2 + 13x + 30

x2 − 4x − 7x + 28

x2 − 11x + 28

x2 − 4x − 7x + 28

x2 − 11x + 28

( 2x2 + 4 ) ( 3x − 5 ) =

( 3x2 − 6x) (4x + 2) =

EXAMPLES

6x3 − 10x2 + 12x− 20

12x3 + 6x2

− 24x2

− 12x

12x3 − 18x2 − 12x12x3 − 18x2 − 12x

Example:

(x +3)(x+1)=(x)(x)+(x)(1)+(3)(x)+(3)((1)

x2 x 3x 3

x2 4x 3

1) Simplify: 5(7n - 2)

Use the distributive property.

5 • 7n

35n - 10- 5 • 2

3(8 12)

4a a 2) Simplify:

6a2 + 9a

3) Simplify: 6rs(r2s - 3) 6rs • r2s

6r3s2 - 18rs

38

4a a

312

4a

- 6rs • 3

4) Simplify: 4t2(3t2 + 2t - 5)

12t4

5) Simplify: - 4m3(-3m - 6n + 4p)

12m4

+ 8t3- 20t2

+ 24m3n - 16m3p

Simplify 4y(3y2 – 1)

1. 7y2 – 1

2. 12y2 – 1

3. 12y3 – 1

4. 12y3 – 4y

Simplify -3x2y3(y2 – x2 + 2xy)

1. -3x2y5 + 3x4y3 – 6x3y4

2. -3x2y6 + 3x4y3 – 6x2y3

3. -3x2y5 + 3x4y3 – 6x2y3

4. 3x2y5 – 3x4y3 + 6x3y4

Try These.

1.) (x+2) (x+8) = x2+10x+16

2.) (x+5) (x-7) = x2-2x-35

3.) (2x+4) (2x-3) = 4x2+2x-12

38

Examples:. Multiply: 2x(3x2 + 2x – 1).

= 6x3 + 4x2 – 2x

= 2x(3x2 ) + 2x(2x) + 2x(–1)

39

Multiply: – 3x2y(5x2 – 2xy + 7y2).

= – 3x2y(5x2 ) – 3x2y(– 2xy) – 3x2y(7y2)

= – 15x4y + 6x3y2 – 21x2y3

40

Example: Multiply: (x – 1)(2x2 + 7x + 3).= (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3)= 2x3 – 2x2 + 7x2 – 7x + 3x – 3= 2x3 + 5x2 – 4x – 3

41

Examples: Multiply: (2x + 1)(7x – 5).

= 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5)

= 14x2 – 10x + 7x – 5

= 14x2 – 3x – 5

First Outer Inner Last

42

Multiply: (5x – 3y)(7x + 6y).

= 35x2 + 30xy – 21yx – 18y2

= 35x2 + 9xy – 18y2

= 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y)First Outer Inner Last

43

(a + b)(a – b)

= a2 – b2

The multiply the sum and difference of two terms, use this pattern:

= a2 – ab + ab – b2

square of the first termsquare of the second term

Special Cases

44

Examples: (3x + 2)(3x – 2)= (3x)2 – (2)2

= 9x2 – 4

(x + 1)(x – 1)= (x)2 – (1)2

= x2 – 1

45

(a + b)2 = (a + b)(a + b)

= a2 + 2ab + b2

= a2 + ab + ab + b2

To square a binomial, use this pattern:

square of the first term

twice the product of the two terms square of the last term

Special Cases

46

Examples: Multiply: (2x – 2)2 .

= (2x)2 + 2(2x)(– 2) + (– 2)2

= 4x2 – 8x + 4

Multiply: (x + 3y)2 .= (x)2 + 2(x)(3y) + (3y)2

= x2 + 6xy + 9y2

Special Cases

FACTORING

GCF

Sum + Product

Factor by Grouping 4 Terms

Special Products

Techniques of Factoring Polynomials

1. Greatest Common Factor (GCF). The GCF for a polynomial is the largest monomial that divides each term of the polynomial.

Factor out the GCF:  23 24 yy

Factoring Polynomials - GCF

23 24 yy

y2

yyy22

Write the two terms in the form of prime factors…

They have in common 2yy

)12(2 2 yy

yy2

1)(2yy

This process is basically the reverse of the distributive property.

50

The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term.

Example: Factor 18x3 + 60x.

GCF = 6x18x3 + 60x = 6x (3x2) + 6x (10) Apply the distributive law

to factor the polynomial.

6x (3x2 + 10) = 6x (3x2) + 6x (10) = 18x3 + 60x

Find the GCF.

= 6x (3x2 + 10)

Factoring - GCF

51

Example: Factor 4x2 – 12x + 20.

Therefore, GCF = 4.

4x2 – 12x + 20 = 4x2 – 4 · 3x + 4 · 5

4(x2 – 3x + 5) = 4x2 – 12x + 20

Check the answer.= 4(x2 – 3x + 5)

Factoring - GCF

52

A common binomial factor can be factored out of certain expressions.

Example: Factor the expression 5(x + 1) – y(x + 1).

5(x + 1) – y(x + 1) = (5 – y)(x + 1)

(5 – y)(x + 1) = 5(x + 1) – y(x + 1)

Check.

Factoring - GCF

Factoring Polynomials - GCF

Factor the GCF:

24233 8124 cabcbaab

3 terms

4ab2( )b - 3a c2 + 2b c2 2

One term

Factoring Polynomials - GCF

)(

EXAMPLE:

)42(3)42(5 xxx

)42( x 5x - 3

Examples

Factor the following polynomial.

)53(4

)53(4

54432012

22

42

xx

xxxx

xxxxxxxx

Examples

Factor the following polynomial.

)15(3

)15(3

353315

42

42

42534253

xyyx

yxyx

yxyxyxyx

57

To factor a trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example,

x2 + 10x + 24 = (x + 4)(x + 6).

Factoring – Sum and Product

4 and 6 add up to 10

4 and 6 multiply to 24

61

65

67

2

2

2

xx

xx

xx

factors of 6 that add up to 7: 6 and 1

1)6( xx

factors of – 6 that add up to – 5: – 6 and 1

factors of – 6 that add up to 1: 3 and – 2

1)6( xx

2)3( xx

Factoring Trinomials

59

Example: Factor x2 – 8x + 15 = (x + a)(x + b)

x2 – 8x + 15 = (x – 3)(x – 5).

Therefore a + b = – 8

= x2 + (a + b)x + ab

It follows that both a and b are negative.

and ab = 15.

Factoring – Sum and Product

60

Example: Factor x2 + 13x + 36. = (x + a)(x + b)

Therefore a and b are two positive factors of 36 whose sum is 13.

x2 + 13x + 36 = (x + 4)(x + 9)

= x2 + (a + b) x + ab

Factoring – Sum and Product

)2()3(

)3(2)3(

623

2

2

23

xx

xxx

xxx There is no GCF for allfour terms.

In this problem we factor GCFby grouping the first two terms and the last two terms.

Factoring 4 Terms by Grouping

62

Some polynomials can be factored by grouping terms to produce a common binomial factor.

= (2x + 3)y – (2x + 3)2

= (2xy + 3y) – (4x + 6) Group terms.

Examples: Factor 2xy + 3y – 4x – 6.

Factor each pair of terms.= (2x + 3)( y – 2) Factor out the common binomial.

Factoring – By Grouping 4 Terms

63

Factor 2a2 + 3bc – 2ab – 3ac.

= 2a2 – 2ab + 3bc – 3ac = (2a2 – 2ab) + (3bc – 3ac)

= 2a(a – b) + 3c(b – a)

Rearrange terms.

Group terms.

Factor.

= 2a(a – b) – 3c(a – b) b – a = – (a – b).

= (2a – 3c)(a – b) Factor.

2a2 + 3bc – 2ab – 3ac

Factoring – By Grouping 4 Terms

Factoring a trinomial when a ≠ 1

Factor 8b2 + 2b – 3

Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring a trinomial when a ≠ 1

Factor 2x2 + 19x - 10 Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring a trinomial when a ≠ 1

Factor 6y2 – 11y - 10 Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring a trinomial when a ≠ 1

Factor 2x2 – x – 3 Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring a trinomial when a ≠ 1

Factor 3t2 + 16t + 5 Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring a trinomial when a ≠ 1

Factor 5x2 + 2x – 3 Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring a trinomial when a ≠ 1

Factor 6b2 – 11b – 2 Multiply 8 -3, and break up the middle term

8b2 + 6b − 4b − 3 Now factor by grouping

8b2 + 6b( ) −4b − 3( )

2b(4b + 3) −1(4b + 3)

(4b + 3)(2b −1)

Factoring the Difference of Two Squares

The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared.

a2 – b2 = (a + b)(a – b) FORMULA:

(a + b)(a – b) = a2– ab + ab – b2 = a2 – b2

Factoring the difference of two squares

Factor x2 – 4y2 Factor 16r2 – 25

(x)2 (2y)2

(x – 2y)(x + 2y)

(4r)2 (5)2

Difference of two squares

DifferenceOf two squares

(4r – 5)(4r + 5)

a2 – b2 = (a + b)(a – b)

Difference of two squares

)4)(4(

)4()(

16

22

2

yy

y

y

Difference of two squares

)95)(95(

)9()5(

8125

22

2

xx

x

x

Difference of two squares

)4)(2)(2(

)4)(4(

)4()(

16

2

22

222

4

yyy

yy

y

y

76

A difference of squares can be factored

using the formula

Example: Factor x2 – 9y2.

= (x)2 – (3y)2

= (x + 3y)(x – 3y)

Write terms as perfect squares.

a2 – b2 = (a + b)(a – b).

Factoring – Special Products