M3 Polynomials - Basic Operations and Factoring.pdf

Recap1: The Absolute Value

If

x

is any real number, the

absolute value

of

x

, written

as

|

x

|

, is defined as:

|

x

|

=







−

x

if

x <

0

0

if

x

= 0

x

if

x >

0

Recap 2: Distance between Real Numbers

The distance between two real numbers

x

and

y

is

d

=

|

x

−

y

|

=

|

y

−

x

|

For

x, y

∈

R

,

Recap 3: Integer Exponents

Let

a

∈

R

, n

∈

Z

. The

nth power of

a

is denoted by

a

.

If

n >

0

, then

a

=

a

·

a

·

a

· · · · ·

a

|

{z

}

n

times

.

If

a

6

= 0

, then

a

= 1

and

a

=

1

a

.

If

n >

0

and

a

6

= 0

, then

a

=

1

a

.

Laws of Exponents

Let

n, m

∈

Z

, a, b

∈

R

, then

1

a

·

a

=

a

2

(

ab

)

=

a

b

3

(

a

)

=

a

4

If

a

6

= 0

, then

a

a

=

a

n−m

5

If

b

6

= 0

, then

Polynomials: Basic

Operations and Factoring

At the end of this lecture, a student must be able to:

Algebraic Expressions and Polynomials

Analgebraic expression is any combination of variables and constants involving a finite number of basic operations.

3x2_−5yz3

√

2x+ 7y

Apolynomial is an algebraic expression that is a sum of constants and/or constants multiplied by variables raised to nonnegative integer exponents.

5x3−2x4y2+y3

Algebraic Expressions and Polynomials

Analgebraic expression is any combination of variables and constants involving a finite number of basic operations.

3x2_−5yz3

√

2x+ 7y

Apolynomial is an algebraic expression that is a sum of constants and/or constants multiplied by variables raised to nonnegative integer exponents.

5x3−2x4y2+y3

Each addend in a polynomial is called aterm of the polynomial. The constant in a term is called its constant coefficient.

Algebraic Expressions and Polynomials

Analgebraic expression is any combination of variables and constants involving a finite number of basic operations.

3x2_−5yz3

√

2x+ 7y

Apolynomial is an algebraic expression that is a sum of constants and/or constants multiplied by variables raised to nonnegative integer exponents.

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 + 6x2−2x 2 Quadratic

4x3_−5y2 _{3 Cubic}

2x3y2−6x4+ 12y 5 5th Degree Polynomial

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 + 6x2−2x 2 Quadratic

4x3_−5y2 _{3 Cubic}

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 + 6x2−2x 2 Quadratic

4x3_−5y2 _{3 Cubic}

2x3_y2_−6x4_{+ 12y 5 5th Degree Polynomial}

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 + 6x2−2x 2 Quadratic

4x3_−5y2 _{3 Cubic}

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 +6x2 −2x 2 Quadratic

4x3_−5y2 _{3 Cubic}

2x3_y2_−6x4_{+ 12y 5 5th Degree Polynomial}

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 + 6x2 −2x 2 Quadratic

Algebraic Expressions and Polynomials

The sum of the exponents of the variables in a term is called its

degree.

5x2y: degree 3

Thedegree of a polynomial is the highest of the degrees of all its terms.

Example Degree

5 0 Constant

2x−5 1 Linear

−3 + 6x2 −2x 2 Quadratic

4x3_−5y2 _{3 Cubic}

2x3_y2_−6x4 _{+ 12y 5 5th Degree Polynomial}

Addition and Subtraction of Polynomials

Like termsare terms that differ only in their constant coefficients.

5x2y,−3yx2

To add or subtract polynomials, combine like terms.

Example:

(4x3_−7x2_{+ 2x−4) + (8x}2_{+ 3x−7)}

= 4x3 _{+ (−7x}2_{+ 8x}2_{) + (2x+ 3x) + ((−4) + (−7))}

Addition and Subtraction of Polynomials

Like termsare terms that differ only in their constant coefficients.

5x2y,−3yx2

To add or subtract polynomials, combine like terms.

Example:

(4x3_−7x2_{+ 2x−4) + (8x}2_{+ 3x−7)}

= 4x3 _{+ (−7x}2_{+ 8x}2_{) + (2x+ 3x) + ((−4) + (−7))}

= 4x3 _+x2_{+ 5x−11}

Addition and Subtraction of Polynomials

Like termsare terms that differ only in their constant coefficients.

5x2y,−3yx2

To add or subtract polynomials, combine like terms.

Example:

(4x3_−7x2_{+ 2x−4) + (8x}2_{+ 3x−7)}

= 4x3 _{+ (−7x}2_{+ 8x}2_{) + (2x+ 3x) + ((−4) + (−7))}

Addition and Subtraction of Polynomials

Like termsare terms that differ only in their constant coefficients.

5x2y,−3yx2

To add or subtract polynomials, combine like terms.

Example:

(4x3_−7x2_{+ 2x−4) + (8x}2_{+ 3x−7)}

= 4x3_{+ (−7x}2_{+ 8x}2_{) + (2x+ 3x) + ((−4) + (−7))}

= 4x3_+x2_{+ 5x−11}

(After combining like terms)

Number of Terms Polynomial Example

1 monomial −7x2

2 binomial 9x4_+x3

(After combining like terms)

Number of Terms Polynomial Example

1 monomial −7x2

2 binomial 9x4_+x3

3 trinomial −17y2+ 11y−60

(After combining like terms)

Number of Terms Polynomial Example

1 monomial −7x2

2 binomial 9x4_+x3

(After combining like terms)

Number of Terms Polynomial Example

1 monomial −7x2

2 binomial 9x4_+x3

3 trinomial −17y2+ 11y−60

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x} = 12y+{−7x−18y+ 15x}

= 12y+ 8x−18y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2]

= 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x} = 12y+{−7x−18y+ 15x}

= 12y+ 8x−18y

= 8x−6y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x} = 12y+{−7x−18y+ 15x}

= 12y+ 8x−18y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x} = 12y+{−7x−18y+ 15x}

= 12y+ 8x−18y

= 8x−6y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x} = 12y+{−7x−18y+ 15x}

= 12y+ 8x−18y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x}

= 12y+{−7x−18y+ 15x} = 12y+ 8x−18y

= 8x−6y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x}

= 12y+ 8x−18y

Multiplication of Polynomials

To multiply polynomials, apply the distributivity of multiplication over addition : a(b+c) =ab+ac, and the laws of exponents. Examples:

2. 8y−3y[4−2(y−1)]

= 8y−3y[4−2y+ 2] = 8y−12y+ 6y2_−6y

= 6y2−10y

3. 12y+{[2(x−3y)−3(3x+ 4y)] + 15x}

= 12y+{[2x−6y−9x−12y] + 15x} = 12y+{−7x−18y+ 15x}

= 12y+ 8x−18y

= 8x−6y

Multiplication of Polynomials

Example:

(6x2−4x+ 1)(4−5x−3x2)

6x2_{−4x+ 1}

× −3x2−5x+ 4

24x2_{−16x+ 4}

−30x3+ 20x2 − 5x

+ −18x4_{+ 12x}3_{− 3x}2

Multiplication of Polynomials

Example:

(6x2−4x+ 1)(4−5x−3x2)

6x2_{−4x+ 1}

× −3x2−5x+ 4

24x2_{−16x+ 4}

−30x3+ 20x2 − 5x

+ −18x4_{+ 12x}3_{− 3x}2

−18x4 −18x3+ 41x2−21x+ 4

Multiplication of Polynomials

Example:

(6x2−4x+ 1)(4−5x−3x2)

6x2_{−4x+ 1}

× −3x2−5x+ 4

24x2_{−16x+ 4}

−30x3+ 20x2 − 5x

+ −18x4_{+ 12x}3_{− 3x}2

Multiplication of Polynomials

Example:

(6x2−4x+ 1)(4−5x−3x2)

6x2_{−4x+ 1}

× −3x2−5x+ 4

24x2_{−16x+ 4}

−30x3+ 20x2 − 5x

+ −18x4_{+ 12x}3_{− 3x}2

−18x4 −18x3+ 41x2−21x+ 4

Multiplication of Polynomials

Example:

(6x2−4x+ 1)(4−5x−3x2)

6x2_{−4x+ 1}

× −3x2−5x+ 4

24x2_{−16x+ 4}

−30x3+ 20x2 − 5x

+ −18x4_{+ 12x}3_{− 3x}2

Special Products

Difference of Two Squares

(x+y)(x−y) = x2−y2

Example:

(9a2_+b)(9a2_{−b) = (9a}2₎2_−(b)2

= 81a4−b2

Special Products

Difference of Two Squares

(x+y)(x−y) = x2−y2

Example:

(9a2_+b)(9a2_−b)

= (9a2₎2_−(b)2

Special Products

Difference of Two Squares

(x+y)(x−y) = x2−y2

Example:

(9a2_+b)(9a2_{−b) = (9a}2₎2_−(b)2

= 81a4−b2

Special Products

Difference of Two Squares

(x+y)(x−y) = x2−y2

Example:

(9a2_+b)(9a2_{−b) = (9a}2₎2_−(b)2

Special Products

Square of a Binomial (Perfect Square Trinomial)

(x±y)2 =x2±2xy+y2

Example:

(2a2₋₁₎2 _{= (2a}2₎2_−(2·2a2_{·1) + (1)}2

= 4a4−4a2+ 1

Special Products

Square of a Binomial (Perfect Square Trinomial)

(x±y)2 =x2±2xy+y2

Example:

(2a2₋₁₎2

= (2a2₎2_−(2·2a2_{·1) + (1)}2

Special Products

Square of a Binomial (Perfect Square Trinomial)

(x±y)2 =x2±2xy+y2

Example:

(2a2₋₁₎2 _{= (2a}2₎2_−(2·2a2_{·1) + (1)}2

= 4a4−4a2+ 1

Special Products

Square of a Binomial (Perfect Square Trinomial)

(x±y)2 =x2±2xy+y2

Example:

(2a2₋₁₎2 _{= (2a}2₎2_−(2·2a2_{·1) + (1)}2

Special Products

Product of Binomials

I

(x+a)(x+b) =x2+ (a+b)x+ab

Example:

(y−9)(y+ 8) = (y)2_{+ ((−9) + 8)y+ (−9)·8}

= y2−y−72

Special Products

Product of Binomials

I

(x+a)(x+b) =x2+ (a+b)x+ab

Example:

(y−9)(y+ 8)

= (y)2_{+ ((−9) + 8)y+ (−9)·8}

Special Products

Product of Binomials

I

(x+a)(x+b) =x2+ (a+b)x+ab

Example:

(y−9)(y+ 8) = (y)2_{+ ((−9) + 8)y+ (−9)·8}

= y2−y−72

Special Products

Product of Binomials

I

(x+a)(x+b) =x2+ (a+b)x+ab

Example:

(y−9)(y+ 8) = (y)2_{+ ((−9) + 8)y+ (−9)·8}

Special Products

Product of Binomials

II

(ax+b)(cx+d) = acx2 + (ad+bc)x+bd

Examples:

(3x+ 5)(2x+ 1) = (3·2)x2_{+ (3·1 + 5·2)x+ 5·1}

= 6x2+ 13x+ 5

Special Products

Product of Binomials

II

(ax+b)(cx+d) = acx2 + (ad+bc)x+bd

Examples:

(3x+ 5)(2x+ 1)

= (3·2)x2_{+ (3·1 + 5·2)x+ 5·1}

Special Products

Product of Binomials

II

(ax+b)(cx+d) = acx2 + (ad+bc)x+bd

Examples:

(3x+ 5)(2x+ 1) = (3·2)x2_{+ (3·1 + 5·2)x+ 5·1}

= 6x2+ 13x+ 5

Special Products

Product of Binomials

II

(ax+b)(cx+d) = acx2 + (ad+bc)x+bd

Examples:

(3x+ 5)(2x+ 1) = (3·2)x2_{+ (3·1 + 5·2)x+ 5·1}

Special Products

Cube of a Binomial

(x±y)3 =x3±3x2y+ 3xy2±y3

Example:

(a−2)3 _{= (a)}3_−3·(a)2_{·2 + 3·a·(2)}2₋₍₂₎3

= a3−6a2+ 12a−8

Special Products

Cube of a Binomial

(x±y)3 =x3±3x2y+ 3xy2±y3

Example:

(a−2)3

= (a)3_−3·(a)2_{·2 + 3·a·(2)}2₋₍₂₎3

Special Products

Cube of a Binomial

(x±y)3 =x3±3x2y+ 3xy2±y3

Example:

(a−2)3 _{= (a)}3_−3·(a)2_{·2 + 3·a·(2)}2₋₍₂₎3

= a3−6a2+ 12a−8

Special Products

Cube of a Binomial

(x±y)3 =x3±3x2y+ 3xy2±y3

Example:

(a−2)3 _{= (a)}3_−3·(a)2_{·2 + 3·a·(2)}2₋₍₂₎3

Special Products

Sum of Two Cubes

(x+y)(x2−xy+y2) = x3+y3

Example: (2a2_{+ 3b)(4a}4 _−6a2_{b+ 9b}2₎

= (2a2_{+ 3b)((2a}2₎2_−2a2_{·3b+ (3b)}2₎

= (2a2)3+ (3b)3 = 8a6 _{+ 27b}3

Special Products

Sum of Two Cubes

(x+y)(x2−xy+y2) = x3+y3

Example: (2a2_{+ 3b)(4a}4_−6a2_{b+ 9b}2₎

= (2a2_{+ 3b)((2a}2₎2_−2a2_{·3b+ (3b)}2₎

Special Products

Sum of Two Cubes

(x+y)(x2−xy+y2) = x3+y3

Example: (2a2_{+ 3b)(4a}4_−6a2_{b+ 9b}2₎

= (2a2_{+ 3b)((2a}2₎2_−2a2_{·3b+ (3b)}2₎

= (2a2)3+ (3b)3 = 8a6 _{+ 27b}3

Special Products

Sum of Two Cubes

(x+y)(x2−xy+y2) = x3+y3

Example: (2a2_{+ 3b)(4a}4_−6a2_{b+ 9b}2₎

= (2a2_{+ 3b)((2a}2₎2_−2a2_{·3b+ (3b)}2₎

= (2a2)3+ (3b)3

Special Products

Sum of Two Cubes

(x+y)(x2−xy+y2) = x3+y3

Example: (2a2_{+ 3b)(4a}4_−6a2_{b+ 9b}2₎

= (2a2_{+ 3b)((2a}2₎2_−2a2_{·3b+ (3b)}2₎

= (2a2)3+ (3b)3 = 8a6_{+ 27b}3

Special Products

Difference of Two Cubes

(x−y)(x2+xy+y2) =x3−y3

Example: (x−2y)(x2_{+ 2xy+ 4y}2₎

= (x−2y)((x)2_{+ (x)(2y) + (2y)}2₎

Special Products

Difference of Two Cubes

(x−y)(x2+xy+y2) =x3−y3

Example: (x−2y)(x2_{+ 2xy+ 4y}2₎

= (x−2y)((x)2_{+ (x)(2y) + (2y)}2₎

= (x)3−(2y)3 = x3_−8y3

Special Products

Difference of Two Cubes

(x−y)(x2+xy+y2) =x3−y3

Example: (x−2y)(x2_{+ 2xy+ 4y}2₎

= (x−2y)((x)2_{+ (x)(2y) + (2y)}2₎

Special Products

Difference of Two Cubes

(x−y)(x2+xy+y2) =x3−y3

Example: (x−2y)(x2_{+ 2xy+ 4y}2₎

= (x−2y)((x)2_{+ (x)(2y) + (2y)}2₎

= (x)3−(2y)3

= x3_−8y3

Special Products

Difference of Two Cubes

(x−y)(x2+xy+y2) =x3−y3

Example: (x−2y)(x2_{+ 2xy+ 4y}2₎

= (x−2y)((x)2_{+ (x)(2y) + (2y)}2₎

Summary of Special Products

1 (x+y)(x−y) = x2 −y2 2 (x±y)2 =x2±2xy+y2

3 (x+a)(x+b) = x2 + (a+b)x+ab 4 (ax+b)(cx+d) =acx2+ (ad+bc)x+bd 5 (x±y)3 =x3±3x2y+ 3xy2±y3

6 (x±y)(x2∓xy+y2) = x3±y3

DO NOT distribute exponents over a SUM: (x+y)n6=xn+yn

Summary of Special Products

1 (x+y)(x−y) = x2 −y2 2 (x±y)2 =x2±2xy+y2

3 (x+a)(x+b) = x2 + (a+b)x+ab 4 (ax+b)(cx+d) =acx2+ (ad+bc)x+bd 5 (x±y)3 =x3±3x2y+ 3xy2±y3

6 (x±y)(x2∓xy+y2) = x3±y3

Division of Polynomials

Monomial Divisor

To divide a polynomial by a monomial, use a+b

d = a d +

b

d. Then,

apply the laws of exponents to simplify.

Example: (6x3y2+ 12x2y3 −9x2y2)÷(3x2y2) 6x3_y2_{+ 12x}2_y3_−9x2_y2

3x2_y2 =

6x3_y2

3x2_y2 +

12x2_y3

3x2_y2 −

9x2_y2

3x2_y2

= 2x+ 4y−3

Division of Polynomials

Monomial Divisor

To divide a polynomial by a monomial, use a+b

d = a d +

b

d. Then,

apply the laws of exponents to simplify.

Example: (6x3y2+ 12x2y3 −9x2y2)÷(3x2y2)

6x3_y2_{+ 12x}2_y3_−9x2_y2

3x2_y2 =

6x3_y2

3x2_y2 +

12x2_y3

3x2_y2 −

9x2_y2

3x2_y2

Division of Polynomials

Monomial Divisor

To divide a polynomial by a monomial, use a+b

d = a d +

b

d. Then,

apply the laws of exponents to simplify.

Example: (6x3y2+ 12x2y3 −9x2y2)÷(3x2y2) 6x3_y2_{+ 12x}2_y3_−9x2_y2

3x2_y2 =

6x3_y2

3x2_y2 +

12x2_y3

3x2_y2 −

9x2_y2

3x2_y2

= 2x+ 4y−3

Division of Polynomials

Monomial Divisor

To divide a polynomial by a monomial, use a+b

d = a d +

b

d. Then,

apply the laws of exponents to simplify.

Example: (6x3y2+ 12x2y3 −9x2y2)÷(3x2y2) 6x3_y2_{+ 12x}2_y3_−9x2_y2

3x2_y2 =

6x3_y2

3x2_y2 +

12x2_y3

3x2_y2 −

9x2_y2

3x2_y2

Long Division

Math 17 (Inst. of Mathematics) Polynomials Lec 3 21 / 31

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x+ 3 4x3 _{−11x+ 20}

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2

2x+ 3 4x3 _{−11x+ 20}

Math 17 (Inst. of Mathematics) Polynomials Lec 3 21 / 31

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2

2x+ 3 4x3 −11x+ 20 −4x3_−6x2

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2

2x+ 3 4x3 _{−11x+ 20}

−4x3−6x2

−6x2 −11x

Math 17 (Inst. of Mathematics) Polynomials Lec 3 21 / 31

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 −3x

2x+ 3 4x3 _{−11x+ 20}

−4x3−6x2

−6x2−11x

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 _−3x

2x+ 3 4x3 −11x+ 20 −4x3_−6x2

−6x2_−11x

6x2 _{+ 9x}

Math 17 (Inst. of Mathematics) Polynomials Lec 3 21 / 31

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 −3x

2x+ 3 4x3 _{−11x+ 20}

−4x3 −6x2

−6x2−11x

6x2 _{+ 9x}

−2x+ 20

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 −3x −1 2x+ 3 4x3 _{−11x+ 20}

−4x3−6x2

−6x2−11x

6x2 _{+ 9x}

−2x+ 20

Math 17 (Inst. of Mathematics) Polynomials Lec 3 21 / 31

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 _{−3x −1}

2x+ 3

4x3 _{−11x+ 20}

−4x3_−6x2

−6x2_−11x

6x2 _{+ 9x}

−2x+ 20 2x + 3

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 −3x −1 2x+ 3 4x3 −11x+ 20

−4x3_−6x2

−6x2−11x

6x2 _{+ 9x}

−2x+ 20 2x + 3 23

Math 17 (Inst. of Mathematics) Polynomials Lec 3 21 / 31

Example:

(4x3_{−11x+20)÷(2x+3)}

= 2x2_−3x−1+ 23

Long Division

2x2 −3x −1 2x+ 3 4x3 −11x+ 20

−4x3_−6x2

−6x2−11x

6x2 _{+ 9x}

−2x+ 20 2x + 3 23

Example:

(4x3_{−11x+20)÷(2x+3) = 2x}2_−3x−1+ 23

Factoring

Similar to prime factorization of integers,

we “break down” polynomials into components that cannot be reduced further

Definition

A polynomial with integer coefficients is said to be

prime if its polynomial factors having integer coefficients are 1, -1, itself and its negative

factored completely(or in completely factored form) if it is expressed as a product of prime polynomials.

Factoring

Similar to prime factorization of integers, we “break down” polynomials into components that cannot be reduced further

Definition

A polynomial with integer coefficients is said to be

prime if its polynomial factors having integer coefficients are 1, -1, itself and its negative

Factoring

Similar to prime factorization of integers, we “break down” polynomials into components that cannot be reduced further

Definition

A polynomial with integer coefficients is said to be

prime if its polynomial factors having integer coefficients are 1, -1, itself and its negative

factored completely(or in completely factored form) if it is expressed as a product of prime polynomials.

Factoring Polynomials

Common Monomial Factor

ax+ay=a(x+y)

Difference of Two Squares

x2−y2 = (x+y)(x−y)

Perfect Square Trinomial

x2±2xy+y2 = (x±y)2

Sum and Difference of Two Cubes

Factoring Polynomials

Common Monomial Factor

ax+ay=a(x+y)

Difference of Two Squares

x2−y2 = (x+y)(x−y)

Perfect Square Trinomial

x2±2xy+y2 = (x±y)2

Sum and Difference of Two Cubes

x3 ±y3 = (x±y)(x2∓xy+y2)

Factoring Polynomials

Common Monomial Factor

ax+ay=a(x+y)

Difference of Two Squares

x2−y2 = (x+y)(x−y)

Perfect Square Trinomial

x2±2xy+y2 = (x±y)2

Sum and Difference of Two Cubes

Factoring Polynomials

Common Monomial Factor

ax+ay=a(x+y)

Difference of Two Squares

x2−y2 = (x+y)(x−y)

Perfect Square Trinomial

x2±2xy+y2 = (x±y)2

Sum and Difference of Two Cubes

x3±y3 = (x±y)(x2∓xy+y2)

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6 = x}2_{+ (2 + 3)x+ 2·3}

= (x+ 2)(x+ 3)

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6 = x}2_{+ (2 + 3)x+ 2·3}

= (x+ 2)(x+ 3)

x2_{−3x−40 = (x−8)(x+ 5)}

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6}

= x2_{+ (2 + 3)x+ 2·3}

= (x+ 2)(x+ 3)

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6 = x}2_{+ (2 + 3)x+ 2·3}

= (x+ 2)(x+ 3)

x2_{−3x−40 = (x−8)(x+ 5)}

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6 = x}2_{+ (2 + 3)x+ 2·3}

= (x+ 2)(x+ 3)

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6 = x}2_{+ (2 + 3)x+ 2·3}

= (x+ 2)(x+ 3)

x2_−3x−40

= (x−8)(x+ 5)

Factoring Polynomials

Trinomials: Case 1

x2+rx+s

Key: If possible, finda and b such that

ab=s a+b=r

Then,x2+rx+s=x2+ (a+b)x+ab= (x+a)(x+b)

Examples:

x2_{+ 5x+ 6 = x}2_{+ (2 + 3)x+ 2·3}

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

3x2+ 8x+ 4 = (3x+ 2)(x+ 2) 2x2_{−7x−15 = (2x+ 3)(x−5)}

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

3x2+ 8x+ 4 = (3x+ 2)(x+ 2) 2x2_{−7x−15 = (2x+ 3)(x−5)}

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

3x2+ 8x+ 4 = (3x+ 2)(x+ 2)

2x2_{−7x−15 = (2x+ 3)(x−5)}

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

Factoring Polynomials

Trinomials: Case 2

rx2+sx+m

Key: If possible, finda, b, c, d such that

ac=r bd=m s= (ad+bc)

Then,rx2+sx+m = acx2+ (ad+bc)x+bd

= (ax+b)(cx+d)

Example:

3x2+ 8x+ 4 = (3x+ 2)(x+ 2) 2x2_{−7x−15 = (2x+ 3)(x−5)}

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by

= x(a+b)−y(a+b) = (x−y)(a+b)

2

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)

−y(a+b) = (x−y)(a+b)

2

xy3+ 2y2−xy−2 = y2(xy+ 2)−(xy+ 2) = (y2−1)(xy+ 2) = (y+ 1)(y−1)(xy+ 2)

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b)

= (x−y)(a+b)

2

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b) = (x−y)(a+b)

2

xy3+ 2y2−xy−2 = y2(xy+ 2)−(xy+ 2) = (y2−1)(xy+ 2) = (y+ 1)(y−1)(xy+ 2)

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b) = (x−y)(a+b)

2

xy3_{+ 2y}2_−xy−2

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b) = (x−y)(a+b)

2

xy3_{+ 2y}2_{−xy−2 = y}2_{(xy+ 2)}

−(xy+ 2) = (y2−1)(xy+ 2) = (y+ 1)(y−1)(xy+ 2)

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b) = (x−y)(a+b)

2

xy3_{+ 2y}2_{−xy−2 = y}2_{(xy+ 2)−(xy+ 2)}

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b) = (x−y)(a+b)

2

xy3_{+ 2y}2_{−xy−2 = y}2_{(xy+ 2)−(xy+ 2)}

= (y2−1)(xy+ 2)

= (y+ 1)(y−1)(xy+ 2)

Factoring by Grouping

Factoring by Grouping

It may be possible to group terms in such a way that each group has a common factor.

Examples:

1

ax+bx−ay−by = x(a+b)−y(a+b) = (x−y)(a+b)

2

Completing the Square

Completing the Square

Add and subtract a perfect square term to obtain a difference of a perfect square trinomial and a perfect square. This will then result in a difference of two squares.

Examples:

1

x4_{+ 4y}4 ₌

x4_{+ 4y}4_{+ 4x}2_y2_−4x2_y2

= (x4+ 4x2y2+ 4y4)−4x2y2

= (x2_{+ 2y}2₎2_−(2xy)2

= (x2_{+ 2y}2_−2xy)(x2_{+ 2y}2_{+ 2xy)}

Completing the Square

Completing the Square

Add and subtract a perfect square term to obtain a difference of a perfect square trinomial and a perfect square. This will then result in a difference of two squares.

Examples:

1

x4_{+ 4y}4 _{= x}4_{+ 4y}4_+4x2_y2_−4x2_y2

= (x4+ 4x2y2+ 4y4)−4x2y2

= (x2_{+ 2y}2₎2_−(2xy)2

Completing the Square

Completing the Square

Add and subtract a perfect square term to obtain a difference of a perfect square trinomial and a perfect square. This will then result in a difference of two squares.

Examples:

1

x4_{+ 4y}4 _{= x}4_{+ 4y}4_{+ 4x}2_y2_−4x2_y2

= (x4_{+ 4x}2_y2_{+ 4y}4_)−4x2_y2

= (x2_{+ 2y}2₎2_−(2xy)2

= (x2_{+ 2y}2_−2xy)(x2_{+ 2y}2_{+ 2xy)}

Completing the Square

Completing the Square

Add and subtract a perfect square term to obtain a difference of a perfect square trinomial and a perfect square. This will then result in a difference of two squares.

Examples:

1

x4_{+ 4y}4 _{= x}4_{+ 4y}4_{+ 4x}2_y2_−4x2_y2

= (x4_{+ 4x}2_y2_{+ 4y}4_)−4x2_y2

= (x2+ 2y2)2−(2xy)2

Completing the Square

Completing the Square

Add and subtract a perfect square term to obtain a difference of a perfect square trinomial and a perfect square. This will then result in a difference of two squares.

Examples:

1

x4_{+ 4y}4 _{= x}4_{+ 4y}4_{+ 4x}2_y2_−4x2_y2

= (x4_{+ 4x}2_y2_{+ 4y}4_)−4x2_y2

= (x2+ 2y2)2−(2xy)2

= (x2_{+ 2y}2_−2xy)(x2_{+ 2y}2_{+ 2xy)}

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4}

= x4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4}

= x4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2)

= (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2)

= (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2)

= (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4}

= (x2_−4)(x2 ₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1) Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1) Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

Factoring Polynomials

Illustration: Factorx4−5x2+ 4.

1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4. 1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Factoring Polynomials

Illustration: Factorx4−5x2+ 4. 1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

= (x−2)(x+ 2)(x+ 1)(x−1)

Both methods lead to the same factored form!

Factoring Polynomials

Illustration: Factorx4−5x2+ 4. 1. Completing the Square

x4_−5x2_{+ 4 = x}4_−4x2_{+ 4−x}2

= (x2₋₂₎2_−x2

= (x2+x−2)(x2−x−2) = (x+ 2)(x−1)(x−2)(x+ 1)

2. Think of this as a trinomial

x4_−5x2_{+ 4 = (x}2_−4)(x2₋₁₎

Recap:

Exercises on Operations:

Perform the following operations:

1 x2+{2x3−[3xy−(4xy+ 7x2)]} 2 4x2y(5x−3xy+ 7y)

3 (2x+y)(x+ 3y) 4 (n3+m2)(n3−m2) 5 (2−r2)(4 + 2r2+r4) 6 (3x+x2)(9x2−3x3+x4) 7 (4x+ 2y)3

Exercises on Factoring:

Factor the following completely:

1 16x2−9y2

2 a2b3c4−a3b4c5+ 2a2b4c4 3 x2(x2−1)−9(x2−1) 4 1 + 8x9

5 8x2 −14x−15 6 x4+ 64

7 x4y8−3x2y4+ 1 8 3b2+ 2ab+ 2ac+ 3bc 9 64n6−16n3+ 1

10 a6−27r3−a4−3a2r−9r2