Math 105-Technical Mathematics I

Multiplying and Factoring Polynomials ( 9 )

Polynomials

I. Polynomials in One Variable

a x a x a x a x an

n

n

n+ + + + +

-

-

1

1

2

2

1 0. . . .

and that n is a nonnegative integer,

and a an,..., 0 are real numbers called coefficients

and an ¹ 0

A. Definition

1. Terms

2. Degree of the polynomial

3. Leading coefficient

4. Constant term

5. Descending order

Examples:

a. 2 8 204 3x x x- + - b. y y2 31

26- +

6. Monomial

7. Binomial

8. Trinomial

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II. Polynomial in Several Variables

A. Definition

1. Degree of a term - sum of the exponents of all the

variables in that term.

2. Degree of a polynomial - the degree of the term

of highest degree.

Examples:

a. 9 12 93 2 4ab a b- +

b. 7 5 3 64 3 3 2 2x y x y x y- + +

III. Expressions That Are Not Polynomials

a. 2 552x xx

- + b. 20 - x c.y

y

+

+

1

73

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IV. Addition and Subtraction of Polynomials

Like terms - terms/expressions that have same variables

raised to the same powers.

Combine/collect like terms

Examples:

a. ( ) ( )- + - + - +5 3 12 7 33 2 3 2x x x x x

b. ( ) ( )8 9 6 32 3 2 3x y xy x y xy- - -

c. ( ) ( )3 2 2 5 8 42 3 2 3x x x x x x- - + - - - +

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Multiplication of Polynomials

A. (binomial)(binomial) : binomial multiplied by binomial,

use FOIL

( )( )x x+ + =4 3 ( ) ( ) ( ) ( )x x x x× + × + × + ×3 4 4 3

F O I L

= + + +x x x2 3 4 12 = x x2 7 12+ +

Examples:

a. ( )( )x x+ - =5 3

b. ( )( )2 3 5a a+ + =

c. ( )( )x y x y2 23 5+ - =

d. ( )4 1 2x + = ( )( )4 1 4 1x x+ + =

e. ( ) ( )( )5 1 5 1 5 12x x x+ = + + =

f. ( ) ( )( )3 2 3 2 3 22y y y- = - - =

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g. ( ) ( )( )2 1 2 1 2 12a a a- = - - =

h. ( )( )2 1 2 1a a- + =

i. ( )( )3 2 3 2y y- + =

j. ( )( )5 1 5 1x x- + =

B. Special Products of Binomials

1. ( )A B A AB B+ = + +2 2 2

2

2. ( )A B A AB B- = - +2 2 2

2

3. ( )( )A B A B A B- + = -2 2

C. Multiplying Two Polynomials

Examples:

a. ( )( )a b a ab b- - + =2 33 2

b. ( )( )4 7 3 2 34 2 2x y x y y y x y- + - =

c. ( )( )2 3 4 2x y x y+ + + =

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Factoring a Monomial from a Polynomial and Factoring byGroupoing

I. Factoring Terms with Common Factors

(Find the greatest common factor)

Examples:

a. 16 12 4 2+ -x x = 4 4 3 2( )+ -x x

b. 14 352 2 3x y x y- =

c. 12 243 2 2x y x y- =

II. Factoring by Grouping

Pairs of terms have a common factor that can be removed in

a process called factoring by grouping.

*Hint: usually for 4 terms

Examples:

a. x x x3 23 5 15+ - - = x x x2 3 5 3( ) ( )+ - + =( )( )x x+ -3 52

b. p p p3 22 9 18- - + =

c. y y y3 23 4 12- - + =

d. 4 12 10 304 2 2x x xa a a+ + + =

e. x ax bx ab2+ + + =

f. x x x3 25 2 10- + - =

g. 10 12 25 302 2m mn mn n- - + =

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Factoring Trinomials

Some trinomials can be factored into the product of two

binomials.

A. To factor a trinomial of the form x bx c2+ + , we look

for two numbers with a product of c and a sum of b.

Examples:

a. z z2 2 24- - = ( )( )z z- +6 4

b. x x2 20- - =

c. x x2 18 81- + =

d. y y2 4 21- - =

e. x y xy2 2 18 64- + =

f. x xy y2 25 6+ + =

g. 10 3 2- -x x =

(hint: factoring -1 from the trinomial)

h. a ab b2 28 33+ - =

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B. Factoring a trinomial of the form ax bx c2+ + , a ¹ 1

Use the trial factor method: use the factor of a and the

factors of c to write all of the possible binomial

factors of the trinomial. Then use FOIL to determine

the correct factorization. To reduce the number of

trial factors that must be considered, remember the

following:

1. Use the sign of the constant term and the

coefficient of x in the trinomial to determine the

signs of the binomial factors. If the constant term

is positive, signs of the binomial factors will be the

same as the sign of the coefficient of x in the tri-

nomial. If the sign of the constant term is negative,

the constant terms in the binomials will have op-

posite signs.

2. If the terms of the trinomial do not have a common

factor, then the terms in either one of the binomial

factors will not have a common factor.

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Examples:

a. Factor: 3 8 42x x- + , + factors of 3 - factors of 4

The terms have no common (coeff. of x2) (constant term)

factor. 1, 3 -1, -4

The constant term is positive. -2, -2

Coefficient of x is negative.

The binomial constant will be

negative.

Write trial factors. Use the Trial Factors Middle Term

Outer and Inner products of ( )( )x x- -1 3 4 - - = -4 3 7x x x

FOIL to determine the middle ( )( )x x- -4 3 1

- - = -x x x12 13

term of the trinomial. ( )( )x x- -2 3 2

- - = -2 6 8x x x

Write the trinomial in factored form3 8 4 2 3 22x x x x- + = - -( )( )

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b. Trinomial of the ax bx x2+ + form can also be factored

by grouping. To factor ax bx c2+ + , first find two

factors a c× whose sum is b. Then use factoring by

grouping to write the factorization of the trinomial.

Examples:

1. Factor: 3 11 82x x+ +

Find two positive factors of 24 + Factors of 24 Sum

(ac = ×3 8) whose sum is 11, the 1, 24 25

coefficient of x. 2, 12 14

3, 8 11

The required sum has been found. The remaining factor need not bechecked. use the factors of 24 3 11 8 3 3 8 82 2x x x x x+ + = + + +

whose sum is 11 to write = + + +( ) ( )3 3 8 82x x x

11x as 3 8x x+ . = 3 1 8 1x x x( ) ( )+ + +

Factor by grouping. = + +( )( )x x1 3 8

Check: ( )( )x x x x x x x+ + = + + + = + +1 3 8 3 8 3 8 3 11 82 2

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2. Factor: 4 17 212z z- -

Find two factors of -84 [ac = × -4 21( )] whose sum is - 17, the

coefficient of z.. Factors of -84 Sum

1, - 84 - 83

- 1, 84 83

2, - 42 - 40

- 2, 42 40

3, - 28 - 25

- 3, 28 25

4, - 21 - 17

(Once the required sum is found, the remaining factors need not bechecked.)

Use the factors of - 84 whose sum is -1 7 to write - 17z as

4 21z z- . Factor by grouping. Recall that - - = - +21 21 21 21z z( )

4 17 21 4 4 21 212 2z z z z z- - = + - -

= + - +( ) ( )4 4 21 212z z z

= + - +4 1 21 1z z z( ) ( )

= + -( )( )z z1 4 21

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Examples:

a. Factor: 6 11 102x x+ -

b. Factor: 12 32 52x x- +

c. Factor: 30 2 4 2y xy x y+ -

d. Factor: 4 15 42x x+ -

e. Factor: y y2 18 72- +

f. Factor: 5 4 2+ -x x

g. Factor: 4 52a a- -

h. Factor: y y y5 38 15- +

i. Factor: 8 8 302x x+ -

j. Factor: 2 5 122x x- -

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Special Factorizations

1. Factors of the Difference of Two Perfect Squares

A B A B A B2 2- = + -( )( )

Examples:

a. x x x2 25 5 5- = + -( )( )

b. 9 252x - =

c. 4 812 2x y- =

d. 25 12x - =

e. x y2 436- =

2. Factors of a Perfect Square Trinomial

A AB B A B2 2 22+ + = +( )

A AB B A B2 2 22- + = -( )

Examples:

a. x x2 8 16+ + =

b. 9 12 42x x+ + =

c. 4 20 252x x- + =

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3. To Factor the sum or the difference of two cubes

A B A B A AB B3 3 2 2+ = + - +( )( )

A B A B A AB B3 3 2 2- = - + +( )( )

Examples:

a. a y a y a y a ay y3 3 3 3 2 264 4 4 4 16+ = + = + - +( ) ( )( )

b. 64 1254y y- =

c. 8 3 3 3x y z+ =

d. x y3 3 1- =

e. ( )x y x+ - =3 3

f. 27 83 6x y- =

g. 8 643 3y x- =

h. ( )x - + =2 643

i. x y3 3+ =

j. x y3 3- =

k. x3 27- =

l. b b3 33- + =( )

m. y x3 9+ =

n. 64 3- =a

o. 5 403 3x y+ =

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Factor Polynomials using a Combination of Techniques

Examples:

a. 3 274 2x x- =

b. 3 24 482 2 2 2x y xy y- + =

c. 24 6 16 42 2x xy xy y- + - =

d. 10 15 202a b ab b- + =

e. 2 544x y xy+ =

f. 6 3 6 92 2x x y- + - =

g. 3 18 27 32 2x x y- + - =

h. 5 455x x- =

i. 2 182( )a b+ - =

j. x3 1

27+ =

k. a b ab3 316- =

l. 3 24 482x x- + =

m 3 12 363 2x x x- - =

n. 4 364 6y x- =

o. x y6 6+ =

p. 6 12 2r s rs+ - =

q. x x y y4 2 2 42 + =

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Polynomial Equations

Definitions

A. An equation is a statement that two expressions are equal.

B. Solving an equation in one variable is to find all the

values of the variable that makes the equation true.

C. Solution set is the set of all solutions of an equation.

D. Equivalent equations are equations that have the same

solution set.

E. Linear equation in one variable: ax b+ = 0, a and

b are real numbers and a ¹ 0.

F. Quadratic equation: ax bx c2 0+ + = , a b c, , are

real numbers and a ¹ 0.

G. Empty set: if equation has no solution, denoted by Æ

II. Equation-Solving Principles

A. If a b= , then ac bc=

B. If a b= , then a c b c+ = +

C. If ab = 0, then a = 0 or b = 0

D. If x k2= , then x k= or x k= -

E. If a b= , then a bn n= , for any positive integer n

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Examples: Solve for the unknown

a. 3 7 2 14 8 1( ) ( )- = - -x x ------>21 6 14 8 8- = - +x x

21 6 22 8- = -x x

21 6 8 22 8 8- + = - +x x x x

21 2 22+ =x

21 21 2 22 21- + = -x

2 1x =

2 2 1 2x ¸ = ¸

x =1

2

b. Solving by factoring:

Example:

1. x x2 5 6 0- - =

( )( )x x- + =6 1 0

so x - =6 0 or x + =1 0

so x = 6 or x = -1

2. x2 9 0- =

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Try to solve: a. 5 5 13- = -x x

b. 7 3 6 11 2( ) ( )x x+ = - +

c. 2 62x x=

d. 5 7+ + =x x

e. x x- + + =3 5 4

f. 10 16 6 02x x- + =

g.3

2

2 4 4

42m m

m

m++ =

-

-

h. 2 5 3 1y y- - - =

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Solving Quadratic Equations by the Quadratic Formula

Consider any quadratic equation in standard form:

ax bx c2 0+ + = , a ¹ 0

xb b ac

a=

- ± -2

4

2

*Discriminant is b ac2 4-

if b ac2 4 0- = --> one real number solution

if b ac2 4 0- > - - > two real-number solutions

if b ac2 4 0- < - - >two imaginary-number

solutions,complex conjugates

Example:

Solve: p p2 4 21 0+ - = using quadratic equation:

a b c= = = -1 4 21, ,

p =- ± - -

=- ± +

=- ±4 4 4 1 21

2 1

4 16 84

2

4 10

2

2 ( )( )

( )=3 or -7

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Example: Solve by quadratic formula

1. 3 5 2 02x x+ - =

2. 2 3 2 02y y- - =

3. 5 3 22m m+ =

4. 3 4 52x x+ =

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Introduction to Polynomial Functions

I. Polynomial Function

A polynomial function P is given by:

P x a x a x a x a x an

n

n

n

n

n( ) .... ,= + + + + +

-

-

-

-

1

1

2

2

1 0

such that the coefficients a a a a an n n, , ,...., ,- -1 2 1 0 are

real numbers and the exponents are whole numbers.

A. Leading coefficients: the first nonzero coefficient ,

an , if the polynomial is in descending order.

B. Degree of the polynomial function is n.

C. Examples of types of polynomial functions and their

names:

Plynomial Function Degree Example

Constant 0 f x( ) = 5

Linear 1 f x x( ) = +3 2

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Quadratic 2 f x x x( ) = - +3 2 32

Cubic 3 f x x x x( ) = + + -3 25 2 4

Examples:

What is the degree of each and every term of the

polynomial and the degree of each polynomial ?

1. f x x x x( ) = + - +3 23 4 2

2. f x x x( ) = + -3 5 34 2

3. f x x y x y xy( ) = + - +2 5 6 23 2 2

II. The Complex Number System