A3 - Polynomials

Polynomials:Basic Operations and Factoring

Mathematics 17

Institute of Mathematics

Lecture 3

Math 17 (Inst. of Mathematics) Polynomials: Basic Operations and Factoring Lec 3 1 / 30

Outline

1 Algebraic Expressions and PolynomialsAddition and Subtraction of PolynomialsMultiplication of PolynomialsDivision of Polynomials

2 FactoringSum and Difference of Two CubesFactoring TrinomialsFactoring By GroupingCompleting the Square


Algebraic Expressions and Polynomials

An algebraic expression is any combination of variables and constantsinvolving a finite number of basic operations.

3x2 − 5yz3√2x+ 7y

A polynomial is an algebraic expression that is a sum of constantsand/or constants multiplied by variables raised to nonnegative integerexponents.

5x3 − 2x4y2 + y3

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




3x2 − 5yz3√2x+ 7y


5x3 − 2x4y2 + y3





3x2 − 5yz3√2x+ 7y


5x3 − 2x4y2 + y3




The sum of the exponents of the variables in a term is called itsdegree.

5x2y

: degree 3

The degree of a polynomial is the highest of the degrees of all itsterms.

Example Degree

5 0 Constant2x− 5 1 Linear

−3 + 6x2 − 2x 2 Quadratic4x3 − 5y2 3 Cubic

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




5x2y : degree 3


Example Degree

5 0 Constant2x− 5 1 Linear






5x2y : degree 3


Example Degree5 0 Constant

2x− 5 1 Linear−3 + 6x2 − 2x 2 Quadratic

4x3 − 5y2 3 Cubic2x3y2 − 6x4 + 12y 5 5th Degree Polynomial




5x2y : degree 3



2x− 5 1 Linear






5x2y : degree 3








5x2y : degree 3




4x3 − 5y2 3 Cubic





5x2y : degree 3






Addition and Subtraction of Polynomials

Like terms are terms that differ only in their constant coefficients.

5x2y,−3yx2

To add or subtract polynomials, combine like terms.

Examples:

1. (4x3 − 7x2 + 2x− 4) + (8x2 + 3x− 7)

= 4x3 + (−7x2 + 8x2) + (2x+ 3x) + ((−4) + (−7))= 4x3 + x2 + 5x− 11




5x2y,−3yx2


Examples:

1. (4x3 − 7x2 + 2x− 4) + (8x2 + 3x− 7)

= 4x3 + (−7x2 + 8x2) + (2x+ 3x) + ((−4) + (−7))= 4x3 + x2 + 5x− 11




5x2y,−3yx2


Examples:

1. (4x3 − 7x2 + 2x− 4) + (8x2 + 3x− 7)

= 4x3 + (−7x2 + 8x2) + (2x+ 3x) + ((−4) + (−7))= 4x3 + x2 + 5x− 11




5x2y,−3yx2


Examples:

1. (4x3 − 7x2 + 2x− 4) + (8x2 + 3x− 7)

= 4x3 + (−7x2 + 8x2) + (2x+ 3x) + ((−4) + (−7))= 4x3 + x2 + 5x− 11




5x2y,−3yx2


Examples:

1. (4x3 − 7x2 + 2x− 4) + (8x2 + 3x− 7)

= 4x3 + (−7x2 + 8x2) + (2x+ 3x) + ((−4) + (−7))

= 4x3 + x2 + 5x− 11




5x2y,−3yx2


Examples:

1. (4x3 − 7x2 + 2x− 4) + (8x2 + 3x− 7)

= 4x3 + (−7x2 + 8x2) + (2x+ 3x) + ((−4) + (−7))= 4x3 + x2 + 5x− 11


(After combining like terms)

Number of Terms Polynomial Example

1 monomial −7x22 binomial 9x4 + x3

3 trinomial −17y2 + 11y − 60



Number of Terms Polynomial Example

1 monomial −7x22 binomial 9x4 + x3

3 trinomial −17y2 + 11y − 60



Number of Terms Polynomial Example1 monomial −7x2

2 binomial 9x4 + x3

3 trinomial −17y2 + 11y − 60



Number of Terms Polynomial Example1 monomial −7x22 binomial 9x4 + x3

3 trinomial −17y2 + 11y − 60



Number of Terms Polynomial Example1 monomial −7x22 binomial 9x4 + x3

3 trinomial −17y2 + 11y − 60


Multiplication of Polynomials

To multiply polynomials, apply the distributivity of · over + :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




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}





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}





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}





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}





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




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




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




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}




Example:(6x2 − 4x+ 1)(4− 5x− 3x2)

6x2 − 4x+ 1× −3x2 − 5x+ 4

24x2 − 16x+ 4−30x3 + 20x2 − 5x

+ −18x4 + 12x3 − 3x2

−18x4 − 18x3 + 41x2 − 21x+ 4



Example:(6x2 − 4x+ 1)(4− 5x− 3x2)

6x2 − 4x+ 1× −3x2 − 5x+ 4

24x2 − 16x+ 4−30x3 + 20x2 − 5x

+ −18x4 + 12x3 − 3x2

−18x4 − 18x3 + 41x2 − 21x+ 4



Example:(6x2 − 4x+ 1)(4− 5x− 3x2)

6x2 − 4x+ 1× −3x2 − 5x+ 4

24x2 − 16x+ 4

−30x3 + 20x2 − 5x+ −18x4 + 12x3 − 3x2

−18x4 − 18x3 + 41x2 − 21x+ 4



Example:(6x2 − 4x+ 1)(4− 5x− 3x2)

6x2 − 4x+ 1× −3x2 − 5x+ 4

24x2 − 16x+ 4−30x3 + 20x2 − 5x

+ −18x4 + 12x3 − 3x2

−18x4 − 18x3 + 41x2 − 21x+ 4



Example:(6x2 − 4x+ 1)(4− 5x− 3x2)

6x2 − 4x+ 1× −3x2 − 5x+ 4

24x2 − 16x+ 4−30x3 + 20x2 − 5x

+ −18x4 + 12x3 − 3x2

−18x4 − 18x3 + 41x2 − 21x+ 4



Example:(6x2 − 4x+ 1)(4− 5x− 3x2)

6x2 − 4x+ 1× −3x2 − 5x+ 4

24x2 − 16x+ 4−30x3 + 20x2 − 5x

+ −18x4 + 12x3 − 3x2

−18x4 − 18x3 + 41x2 − 21x+ 4


Special Products

Product of a Sum and a Difference

(x+ y)(x− y) = x2 − y2

... is a Difference of Two Squares

Example:(9a2 + b)(9a2 − b) = (9a2)2 − (b)2

= 81a4 − b2


Special Products


(x+ y)(x− y) = x2 − y2


Example:(9a2 + b)(9a2 − b) = (9a2)2 − (b)2

= 81a4 − b2


Special Products


(x+ y)(x− y) = x2 − y2


Example:(9a2 + b)(9a2 − b)

= (9a2)2 − (b)2

= 81a4 − b2


Special Products


(x+ y)(x− y) = x2 − y2


Example:(9a2 + b)(9a2 − b) = (9a2)2 − (b)2

= 81a4 − b2


Special Products


(x+ y)(x− y) = x2 − y2


Example:(9a2 + b)(9a2 − b) = (9a2)2 − (b)2

= 81a4 − b2


Special Products

Square of a Binomial

(x± y)2 = x2 ± 2xy + y2

... is a Perfect Square Trinomial.

Example:(2a2 − 1)2 = (2a2)2 − 2 · 2a2 · 1 + (1)2

= 4a4 − 4a2 + 1


Special Products


(x± y)2 = x2 ± 2xy + y2


Example:(2a2 − 1)2 = (2a2)2 − 2 · 2a2 · 1 + (1)2

= 4a4 − 4a2 + 1


Special Products


(x± y)2 = x2 ± 2xy + y2


Example:(2a2 − 1)2

= (2a2)2 − 2 · 2a2 · 1 + (1)2

= 4a4 − 4a2 + 1


Special Products


(x± y)2 = x2 ± 2xy + y2


Example:(2a2 − 1)2 = (2a2)2 − 2 · 2a2 · 1 + (1)2

= 4a4 − 4a2 + 1


Special Products


(x± y)2 = x2 ± 2xy + y2


Example:(2a2 − 1)2 = (2a2)2 − 2 · 2a2 · 1 + (1)2

= 4a4 − 4a2 + 1


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


(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


(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


(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 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



Examples:

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

= (3 · 2)x2 + (3 · 1 + 5 · 2)x+ 5 · 1= 6x2 + 13x+ 5


Special Products



Examples:

(3x+ 5)(2x+ 1) = (3 · 2)x2 + (3 · 1 + 5 · 2)x+ 5 · 1

= 6x2 + 13x+ 5


Special Products



Examples:

(3x+ 5)(2x+ 1) = (3 · 2)x2 + (3 · 1 + 5 · 2)x+ 5 · 1= 6x2 + 13x+ 5


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

= 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

= 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

= a3 − 6a2 + 12a− 8


Special Products

Sum of Two Cubes

(x+ y)(x2 − xy + y2) = x3 + y3

Example: (2a2 + 3b)(4a4 − 6a2b+ 9b2)

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

= 8a6 + 27b3


Special Products

Sum of Two Cubes

(x+ y)(x2 − xy + y2) = x3 + y3

Example: (2a2 + 3b)(4a4 − 6a2b+ 9b2)

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

= 8a6 + 27b3


Special Products

Sum of Two Cubes

(x+ y)(x2 − xy + y2) = x3 + y3

Example: (2a2 + 3b)(4a4 − 6a2b+ 9b2)

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

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

= 8a6 + 27b3


Special Products

Sum of Two Cubes

(x+ y)(x2 − xy + y2) = x3 + y3

Example: (2a2 + 3b)(4a4 − 6a2b+ 9b2)

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

= 8a6 + 27b3


Special Products

Sum of Two Cubes

(x+ y)(x2 − xy + y2) = x3 + y3

Example: (2a2 + 3b)(4a4 − 6a2b+ 9b2)

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

= 8a6 + 27b3


Special Products

Difference of Two Cubes

(x− y)(x2 + xy + y2) = x3 − y3

Example: (x− 2y)(x2 + 2xy + 4y2)

= (x− 2y)((x)2 + (x)(2y) + (2y)2)= (x)3 + (2y)3

= x3 + 8y3


Special Products


(x− y)(x2 + xy + y2) = x3 − y3


= (x− 2y)((x)2 + (x)(2y) + (2y)2)= (x)3 + (2y)3

= x3 + 8y3


Special Products


(x− y)(x2 + xy + y2) = x3 − y3


= (x− 2y)((x)2 + (x)(2y) + (2y)2)

= (x)3 + (2y)3

= x3 + 8y3


Special Products


(x− y)(x2 + xy + y2) = x3 − y3


= (x− 2y)((x)2 + (x)(2y) + (2y)2)= (x)3 + (2y)3

= x3 + 8y3


Special Products


(x− y)(x2 + xy + y2) = x3 − y3


= (x− 2y)((x)2 + (x)(2y) + (2y)2)= (x)3 + (2y)3

= x3 + 8y3


Special Products

(x+ y)(x− y) = x2 − y2

(x± y)2 = x2 ± 2xy + y2

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


(x± y)3 = x3 ± 3x2y + 3xy2 ± y3

(x± y)(x2 ∓ xy + y2) = x3 ± y3

DO NOT distribute exponents over a SUM: (x+ y)n 6= xn + yn


Special Products

(x+ y)(x− y) = x2 − y2

(x± y)2 = x2 ± 2xy + y2

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


(x± y)3 = x3 ± 3x2y + 3xy2 ± y3

(x± y)(x2 ∓ xy + y2) = x3 ± y3

DO NOT distribute exponents over a SUM: (x+ y)n 6= xn + yn


Division of Polynomials

Monomial Divisor

To divide a polynomial by a monomial, usea+ b

d=

a

d+

b

d. Then, apply

the laws of exponents to simplify.

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

6x3y2 + 12x2y3 − 9x2y2

3x2y2=

6x3y2

3x2y2+

12x2y3

3x2y2− 9x2y2

3x2y2

= 2x+ 4y − 3



Monomial Divisor


d=

a

d+

b

d. Then, apply



6x3y2 + 12x2y3 − 9x2y2

3x2y2=

6x3y2

3x2y2+

12x2y3

3x2y2− 9x2y2

3x2y2

= 2x+ 4y − 3



Monomial Divisor


d=

a

d+

b

d. Then, apply



6x3y2 + 12x2y3 − 9x2y2

3x2y2

=6x3y2

3x2y2+

12x2y3

3x2y2− 9x2y2

3x2y2

= 2x+ 4y − 3



Monomial Divisor


d=

a

d+

b

d. Then, apply



6x3y2 + 12x2y3 − 9x2y2

3x2y2=

6x3y2

3x2y2+

12x2y3

3x2y2− 9x2y2

3x2y2

= 2x+ 4y − 3



Monomial Divisor


d=

a

d+

b

d. Then, apply



6x3y2 + 12x2y3 − 9x2y2

3x2y2=

6x3y2

3x2y2+

12x2y3

3x2y2− 9x2y2

3x2y2

= 2x+ 4y − 3


Long Division


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x+ 3)

4x3 − 11x+ 20


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2

2x+ 3)

4x3 − 11x+ 20


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x6x2 + 9x


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x6x2 + 9x

− 2x+ 20


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x − 1

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x6x2 + 9x

− 2x+ 20


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x − 1

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x6x2 + 9x

− 2x+ 202x + 3


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x − 1

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x6x2 + 9x

− 2x+ 202x + 3

23


Example:

(4x3 − 12x+ 20)÷ (2x+ 3)

= 2x2 − 3x− 1 +23

2x+ 3

Long Division

2x2 − 3x − 1

2x+ 3)

4x3 − 11x+ 20− 4x3 − 6x2

− 6x2 − 11x6x2 + 9x

− 2x+ 202x + 3

23


Example:

(4x3 − 12x+ 20)÷ (2x+ 3) = 2x2 − 3x− 1 +23

2x+ 3

Factoring

- similar to prime factorization of integers

- “breaking down” polynomials into components that cannot be reducedfurther


Factoring

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 expressedas a product of prime polynomials.


Common Monomial Factor

ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)

[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2

= 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)

[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2

(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)

[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)

[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4)

= (x+ 1)

[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)

[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)[x(x+ 1)− (3x+ 4)]

= (x+ 1)[x2 − 2x− 4]



ax+ ay = a(x+ y)

Examples:

1

6x5y4 − 14x9y2 = 2x5y2(3y2 − 7x4)

2

x(x+ 1)2 − (x+ 1)(3x+ 4) = (x+ 1)[x(x+ 1)− (3x+ 4)]= (x+ 1)[x2 − 2x− 4]


Difference of Two Squares

x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2

= (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2

= (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12

= (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)

= (4m2 + n6)(2m+ n3)(2m− n3)



x2 − y2 = (x+ y)(x− y)

Examples:

1

4a2 − b2 = (2a)2 − (b)2 = (2a+ b)(2a− b)

2

16m4 − n12 = (4m2)2 − (n6)2

= (4m2 + n6)(4m2 − n6)= (4m2 + n6)(2m+ n3)(2m− n3)


Perfect Square Trinomial

x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2 = (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8 = (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2



x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2

= (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8 = (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2



x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2 = (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8 = (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2



x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2 = (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8 = (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2



x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2 = (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8

= (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2



x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2 = (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8 = (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2



x2 ± 2xy + y2 = (x± y)2

Examples:

1

4a2 + 4ab+ b2 = (2a)2 + 2 · 2a · b+ b2

= (2a+ b)2

2

121m4 − 44m2n4 + 4n8 = (11m2)2 − 2 · 11m2 · 2n4 + (2n4)2

= (11m2 − 2n4)2


Factoring Polynomials

Sum and Difference of Two Cubes

x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125

= (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)

(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3

= (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)

(9 + 12m+ 16m2)




x3 ± y3 = (x± y)(x2 ∓ xy + y2)

Examples:z3 + 125 = (z)3 + (5)3

= (z + 5)(z2 − 5z + 25)

27− 64m3 = (3)3 − (4m)3

= (3−m)(9 + 12m+ 16m2)



Trinomials: Case 1

x2 + rx+ s

Key: If possible, find a 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)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6 = x2 + (2 + 3)x+ 2 · 3

= (x+ 2)(x+ 3)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6

= x2 + (2 + 3)x+ 2 · 3= (x+ 2)(x+ 3)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6 = x2 + (2 + 3)x+ 2 · 3

= (x+ 2)(x+ 3)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6 = x2 + (2 + 3)x+ 2 · 3

= (x+ 2)(x+ 3)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6 = x2 + (2 + 3)x+ 2 · 3

= (x+ 2)(x+ 3)

x2 − 3x− 40

= x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6 = x2 + (2 + 3)x+ 2 · 3

= (x+ 2)(x+ 3)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5

= (x− 8)(x+ 5)



Trinomials: Case 1

x2 + rx+ s


ab = s a+ b = r


= (x+ a)(x+ b)

Examples:x2 + 5x+ 6 = x2 + (2 + 3)x+ 2 · 3

= (x+ 2)(x+ 3)

x2 − 3x− 40 = x2 + [(−8) + 5]x+ (−8) · 5= (x− 8)(x+ 5)



Trinomials: Case 2

rx2 + sx+m

Key: If possible, find a, 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 = 3 · 1x2 + (3 · 2 + 2 · 1)x+ 2 · 2= (3x+ 2)(x+ 2)



Trinomials: Case 2

rx2 + sx+m




= (ax+ b)(cx+ d)

Example:

3x2 + 8x+ 4 = 3 · 1x2 + (3 · 2 + 2 · 1)x+ 2 · 2= (3x+ 2)(x+ 2)



Trinomials: Case 2

rx2 + sx+m




= (ax+ b)(cx+ d)

Example:

3x2 + 8x+ 4 = 3 · 1x2 + (3 · 2 + 2 · 1)x+ 2 · 2= (3x+ 2)(x+ 2)



Trinomials: Case 2

rx2 + sx+m




= (ax+ b)(cx+ d)

Example:

3x2 + 8x+ 4

= 3 · 1x2 + (3 · 2 + 2 · 1)x+ 2 · 2= (3x+ 2)(x+ 2)



Trinomials: Case 2

rx2 + sx+m




= (ax+ b)(cx+ d)

Example:

3x2 + 8x+ 4 = 3 · 1x2 + (3 · 2 + 2 · 1)x+ 2 · 2

= (3x+ 2)(x+ 2)



Trinomials: Case 2

rx2 + sx+m




= (ax+ b)(cx+ d)

Example:

3x2 + 8x+ 4 = 3 · 1x2 + (3 · 2 + 2 · 1)x+ 2 · 2= (3x+ 2)(x+ 2)



Factoring by Grouping

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

Examples:

ax+ bx− ay − by

= x(a+ b)− y(a+ b)= (x− y)(a+ b)

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





Examples:

ax+ bx− ay − by = x(a+ b)

− y(a+ b)= (x− y)(a+ b)






Examples:

ax+ bx− ay − by = x(a+ b)− y(a+ b)

= (x− y)(a+ b)






Examples:

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






Examples:


xy3 + 2y2 − xy − 2

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





Examples:


xy3 + 2y2 − xy − 2 = y2(xy + 2)

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





Examples:


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

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





Examples:


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

= (y + 1)(y − 1)(xy + 2)





Examples:





Completing the Square

Add and subtract a Perfect Square term to obtain a difference of a PerfectSquare Trinomial and a Perfect Square. This will then result in aDifference of Two Squares.

Example:

x4 + 4y4 =

x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)





Example:

x4 + 4y4 =

x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)





Example:

x4 + 4y4 =

x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)





Example:

x4 + 4y4 = x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)





Example:

x4 + 4y4 = x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)





Example:

x4 + 4y4 = x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)





Example:

x4 + 4y4 = x4 + 4y4 + 4x2y2 − 4x2y2

= (x4 + 4x2y2 + 4y4)− 4x2y2

= (x2 + 2y2)2 − (2xy)2

= (x2 + 2y2 − 2xy)(x2 + 2y2 + 2xy)



Illustration: Factor x4 − 5x2 + 4.

1. Completing the Square

x4 − 5x2 + 4

= x4 − 4x2 + 4− x2

= (x− 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!





x4 − 5x2 + 4

= x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

= (x2 + x− 2)(x2 − x− 2)

= (x+ 2)(x− 1)(x− 2)(x+ 1)


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

= (x2 + x− 2)(x2 − x− 2)

= (x+ 2)(x− 1)(x− 2)(x+ 1)


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

= (x2 + x− 2)(x2 − x− 2)

= (x+ 2)(x− 1)(x− 2)(x+ 1)


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4

= (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)

= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)

= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)

= (x− 2)(x+ 2)(x+ 1)(x− 1)






x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)




Illustration: Factor x4 − 5x2 + 4.1. Completing the Square

x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)





x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)





x4 − 5x2 + 4 = x4 − 4x2 + 4− x2

= (x− 2)2 − x2

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


x4 − 5x2 + 4 = (x2 − 4)(x2 − 1)= (x− 2)(x+ 2)(x+ 1)(x− 1)



Exercises:

1 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 (14a3b3c3 +35a2b4c3)÷ 7ab3c3

8 (x5 − 2x4 + 3x3 − 2x2 + 2x)÷(x2 − 2x+ 2)

2 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 + 647 x4y8 − 3x2y4 + 18 3b2 + 2ab+ 2ac+ 3bc9 64n6 − 16n3 + 110 a6 − 27r3 − a4 − 3a2r − 9r2


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A3 - Polynomials