Section 5.1 Polynomials Addition And Subtraction

Section 5.1

Polynomials Addition And Subtraction

OBJECTIVES

A Classify polynomials.

OBJECTIVES

B Find the degree of a polynomial and write descending order.

OBJECTIVES

C Evaluate a polynomial.

OBJECTIVES

D Add or subtract polynomials.

OBJECTIVES

E Solve applications involving sums or differences of polynomials.

DEFINITION

Degree of a Polynomial in One Variable

The degree of a polynomial in one variable is the greatest exponent of that variable.

DEFINITION

Degree of a Polynomial in Several Variables

The greatest sum of the exponents of the variables in any one term of the polynomial.

RULES

Properties for Adding Polynomials

P + Q = Q + P If P , Q, and R are polynomials,

Commutative Property

RULES

If P , Q, and R are polynomials,

P + Q + R = P + Q + R

Associative Property

Properties for Adding Polynomials

RULES

If P , Q, and R are polynomials,

P Q + R = PQ + PR

Q + R P = QP + RP

Properties for Adding Polynomials

RULES

Subtracting Polynomials

a – (b + c) = a – b – c

Section 5.1A,B

Exercise #1

Chapter 5

Classify as a monomial, binomial, or trinomial and give the degree.

xy3z4 – x7

Binomial.

Degree is determined by comparing

Degree 1st term: x1 y3 z4

1 + 3 + 4 = 8

Degree 2nd term: x7 7

Degree 8

Section 5.1D

Exercise #5

Chapter 5

Add 6x3 + 8x2 – 6x – 4 and 6 – 3x + x2 – 3x3 .

6x3 + 8x2 – 6x – 4

– 3x3 + x2 – 3x + 6

3x3

METHOD 1

+ 9x2 – 9x + 2

3 2 2 36 + 8 – 6 – 4 6 – 3 + – 3x x x + x x x

= 6x3 + 8x2 – 6x – 4 + 6 – 3x + x2 – 3x3

= 3x3 + 9x2 – 9x + 2

Add 6x3 + 8x2 – 6x – 4 and 6 – 3x + x2 – 3x3 .

= 6x3 – 3x3 + 8x2 + x2 – 6x – 3x – 4 + 6

METHOD 2

Section 5.1D

Exercise #6

Chapter 5

Subtract 8x3 – 6x2 + 5x – 3 from 5x3 + 3x2 + 3.

5x3 + 3x 2 + 3 –

METHOD 1

8x3 – 6x2 + 5x – 3

Subtract 8x3 – 6x2 + 5x – 3 from 5x3 + 3x2 + 3.

5x3 + 3x 2 + 3 –

5x3 + 3x 2 + 3

METHOD 1

8x3 – 6x2 + 5x – 3

+

Subtract 8x3 – 6x2 + 5x – 3 from 5x3 + 3x2 + 3.

5x3 + 3x 2 + 3 –

– 8x3 + 6x2 – 5x + 3

– 3x3

5x3 + 3x 2 + 3

METHOD 1

8x3 – 6x2 + 5x – 3

+ 9x2 – 5x + 6

+

3 2 3 25 + 3 + 3 – 8 – 6 + 5 – 3x x x x x

= – 3x3 + 9x2 – 5x + 6

= 5x3 + 3x 2 + 3 – 8x3 + 6x 2 – 5x + 3

Subtract 8x3 – 6x2 + 5x – 3 from 5x3 + 3x2 + 3.

= 5x3 – 8x3 + 3x2 + 6x2 – 5x + 3 + 3

METHOD 2

Section 5.2

Multiplication of Polynomials

OBJECTIVES

A Multiply a monomial by a polynomial.

OBJECTIVES

B Multiply two polynomials.

OBJECTIVES

C Use the FOIL method to multiply two binomials.

OBJECTIVES

D Square a binomial sum or difference.

OBJECTIVES

E Find the product of the sum and the difference of two terms.

OBJECTIVES

F Use the ideas discussed to solve applications.

RULES

Multiplication of Polynomials

If P , Q, and R are polynomials,

P Q = Q P P Q R = P Q R

To Multiply Two Binomials

USING FOIL

= x 2 + (b + a)x + ab

(x + a)(x + b)

= x 2 + bx + ax + ab First Outside Inside Last

RULETo Square a Binomial Sum

(x + a)2

= (x + a)(x + a)

= x 2 + 2ax + a2

RULETo Square a Binomial Difference

(x – a)2

= (x – a)(x – a)

= x 2 – 2ax + a2

Sum and Difference of Same Two Monomials

PROCEDURE

(x + a)(x – a) = x 2 – a2

(x – a)(x + a) = x 2 – a2

Section 5.2B,C

Exercise #8a

Chapter 5

2

M

ultip

– 2 – 4 – 5

ly.

x x x

x2 – 4x – 5

x3 – 4x2 – 5x

x3 – 6x2 + 3x + 10

x – 2

– 2x2 + 8x + 10

METHOD 1

2 2= – 4 – 5 + – 2 – 4 – 5x x x x x

= x3 – 4x2 – 5x – 2x2 + 8x + 10

= x3 – 4x2 – 2x2 – 5x + 8x + 10

= x3 – 6x2 + 3x + 10

2

M

ultip

– 2 – 4 – 5

ly.

x x x

METHOD 2

Section 5.2D

Exercise #9b

Chapter 5

Multiply.

= 9x2

23 – 4x y

2 2= 3 – 2(3 )(4 ) + 4x x y y

2 2 2 – = – 2 + x a x ax a

– 24xy + 16y2

Section 5.2E

Exercise #10

Chapter 5

Multiply.

2 2= 3 – 4x y

= 9x2 – 16y2

3 + 4 3 – 4x y x y

2 2 + – = – x a x a x a

Product of Sum and Difference ofSame Two Monomials

Section 5.3

The Greatest Common Factor and Factoring by Grouping

OBJECTIVES

A Factor out the greatest common factor in a polynomial.

OBJECTIVES

B Factor a polynomial with four terms by grouping.

GREATEST COMMON FACTOR

is the Greatest Common monomial Factor (GCF) of a polynomial in x if:

axn

1. a is the greatest integer that divides each coefficient.

GREATEST COMMON FACTOR

is the Greatest Common monomial Factor (GCF) of a polynomial in x if:

axn

2. n is the smallest exponent of x in all the terms.

PROCEDURE

Factoring by Grouping

1. Group terms with common factors using the associative property.

PROCEDURE

2. Factor each resulting binomial.

Factoring by Grouping

PROCEDURE

3. Factor out the binomial using the GCF, by the distributive property.

Factoring by Grouping

Section 5.3B

Exercise #12

Chapter 5

Factor completely.

2 5 3 2 = 3 2 + 2 + 5 + 5x x x x

6x7 + 6x5 + 15x4 + 15x2

2 3 2 2 = 3 2 + 1 + 5 + 1 x x x x

2 2 3 = 3 + 1 2 + 5 x x x

2 2 3 = 3 + 1 2 + 5x x x

Section 5.4

Factoring Trinomials

OBJECTIVES

A Factor a trinomial of the form . x 2 + bx + c

OBJECTIVES

B Factor a trinomial of the form using trial and error.

ax 2 + bx + c

OBJECTIVES

C Factor a trinomial of the form using the ac test.

ax 2 + bx + c

PROCEDURE

Factoring Trinomials

x2 + (b + a)x + ab

= (x + a)(x + b)

RULE

The ac Test

is factorable only if there are two integers whose product is ac and sum is b.

ax 2 + bx + c

Section 5.4A,B,C

Exercise #13b

Chapter 5

Factor completely.

= 2 + 5 – 2x y x y

2x2 + xy – 10y2The ac Method Find factors of ac (–20) whose sum is (1) and replace the middle term (xy).

= 2x2 – 4xy + 5xy – 10y2

= (2x 2 – 4xy) + (5xy – 10y 2)

= 2x(x – 2y) + 5y(x – 2y)

Section 5.5

Special Factoring

OBJECTIVES

A Factor a perfect square trinomial.

OBJECTIVES

B Factor the difference of two squares.

OBJECTIVES

C Factor the sum or difference of two cubes.

PROCEDURE

Factoring Perfect Square Trinomials

x 2 + 2 ax + a2 = (x + a)2

x 2 – 2 ax + a2 = (x – a)2

PROCEDURE

Factoring the Difference of Two Squares

x2 – a2 = (x + a)(x – a)

PROCEDURE

Factoring the Sum and Difference of Two Cubes

x3+ a3 = (x + a)(x 2 – ax + a2 )

x3 – a3 = (x – a)(x 2+ ax + a2 )

Section 5.5A

Exercise #15a

Chapter 5

Factor completely.

16x2 – 24xy + 9y2

Perfect Square Trinomial

2 = 4 – 3x y

22 2 – 2 + = – x ax a x a

2 2 = 16 – 2 12 + 9x xy y

Section 5.5

Exercise #16

Chapter 5

Factor completely.

22 2 2= + 4 – 4x y x y

x4 – 16y4

Difference of Two Squares

Factor x2 – 4y2

2 2= + 4 + 2 – 2x y x y x y

2 2 – = + – x a x a x a

2 22 2 = – 4x y

Section 5.5B

Exercise #17

Chapter 5

Factor completely.

x2 – 10x + 25 – y2

Perfect Square Trinomial

2 2 = – 5 – x y

= – 5 + – 5 – x y x y

= + – 5 – 5 x y x – y

x2 – 2ax + a2 = x – a 2

Difference of Two Squares

2 2 – = + – x a x a x a

Section 5.5c

Exercise #18a

Chapter 5

Factor completely.

2 = 3 + 2 9 x y x

27x3 + 8y3

3 3 = 3 + 2x y

Sum of Two Cubes

3 3 2 2 + = + – + x a x a x ax a

– 6xy + 4y2

Section 5.6

General Methods of Factoring

OBJECTIVES

A Factor a polynomial using the procedure given in the text.

PROCEDUREA General Factoring Strategy

1. Factor out the GCF, if there is one.

2. Look at the number of terms in the given polynomial.

PROCEDUREA General Factoring Strategy

If there are two terms, look for:

Difference of Two Squaresx 2 – a2 = (x + a)(x – a)

PROCEDUREA General Factoring Strategy

If there are two terms, look for:

Difference of Two Cubesx3 – a3 = (x – a)(x 2

+ ax + a2 )

PROCEDUREA General Factoring Strategy

If there are two terms, look for:

Sum of Two Cubesx3+ a3 = (x + a)(x 2 – ax + a2 )

PROCEDUREA General Factoring Strategy

If there are two terms, look for:

The sum of two squares, is not factorable. x 2 + a2

PROCEDUREA General Factoring Strategy

If there are three terms, look for:

Perfect square trinomial

x 2 + 2ab + b2 = (x + a)2

x 2 – 2ab + b2 = (x – a)2

PROCEDUREA General Factoring Strategy

If there are three terms, look for:

Trinomials of the form

ax 2 + bx + c (a > 0)

PROCEDUREA General Factoring Strategy

If a < 0, factor out – 1 first.

Use the ac method or trial and error.

PROCEDUREA General Factoring Strategy

If there are four terms:

Factor by grouping.

PROCEDUREA General Factoring Strategy

3. Check the result by multiplying the factors.

Section 5.6A

Exercise #20b

Chapter 5

Factor completely.

2 2 2 = 3 16 – 24 + 9y x xy y

48x2y2 – 72xy3 + 27y4

22 = 3 4 – 3y x y

Perfect Square Trinomial

22 2 – 2 + = – x ax a x a

2 22 = 3 4 – 2 12 + 3 y x xy y

Section 5.6A

Exercise #21

Chapter 5

Factor completely: 9x3y – 33x2y2 – 12xy3

= 3xy [3x2 – 12xy + xy – 4y2]

The ac Method Find factors of ac (–12) whose sum is (–11) and replace the middle term (–11xy).

2 2 = 3 3 – 12 + – 4xy x xy xy y [ ] = 3 3 – 4 + – 4xy x x y y x y [ ]

= 3 – 4 3 + xy x y x y [ ]

= 3xy [3x2 – 11xy – 4y2]

Factor completely: 9x3y – 33x2y2 – 12xy3

= 3xy [3x2 – 12xy + xy – 4y2]

= 3 3 + – 4xy x y x y

The ac Method Find factors of ac (–12) whose sum is (–11) and replace the middle term (–11xy).

2 2 = 3 3 – 12 + – 4xy x xy xy y [ ] = 3 3 – 4 + – 4xy x x y y x y [ ]

= 3xy [3x2 – 11xy – 4y2]

Section 5.6A

Exercise #22

Chapter 5

Factor completely.

16x3 – 12x2 – 4x + 3

2 = 4 – 3 4 – 1x x

= 4 – 3 2 + 1 2 – 1x x x

2 = 4 4 – 3x x

Difference of Two Squares

2 2 – = + – x a x a x a

– 1 4 – 3x

Section 5.7

Solving Equations by Factoring: Applications

OBJECTIVES

A Solve equations by factoring.

OBJECTIVES

B Use Pythagorean theorem to find the length of one side of a right triangle when the lengths of the other two sides are given.

OBJECTIVES

C Solve applications involving quadratic equations.

PROCEDURE

1. Set equation equal to 0.

2. Factor Completely.

3. Set each linear Factor equal to 0 and solve each.

O

F

F

DEFINITION

Pythagorean Theorem

In symbols,c2 = a2 + b2

a

b

c

Section 5.7A

Exercise #23b

Chapter 5

Solve. 6x2 + 7x = 3

3 – 1 2 + 3 = 0x x

3x = 1

6x2 + 7x – 3 = 0

x =

13

, –32

3x – 1 = 0 2x + 3 = 0or

2x = – 3

x =

13

x = – 32

or

O

F

F