Adding and Subtracting PolynomialsLesson 8-1

Vocabulary• Monomial: a real number, or variable with a whole

number as an exponent:• Binomial: two monomials being added or subtracted.• Trinomial: three monomials being added or

subtracted.

Label each as a monomial, binomial or trinomial.

5x2 6x2 + 4y7 + -2w3 mn2 – 7y

Degree of a monomial: the SUM of the exponents of its variables.

Problem 1What is the degree of each monomial:

1. 5x

2. 6x3y2

3. 4

Problem 2What is the sum or difference?

1. 3x2 + 5x2 – x2

2. 4x3y – x3y + 6x3y

VocabularyPolynomial: a monomial or sum or monomial

Degree of the polynomial: the variable with the largest exponent is also the degree

3x4 + 5x2 – 7x + 1

The degree is 4.

Standard form is that you place the monomials in descending order from left to right.

Polynomial Degree

Name # of terms Name Using # of

Terms 6 0 Constant 0 Monomial

5x + 1 1 Linear 1 Binomial4x2 + 7x – 3 2 Quadratic 2 Trinomial

2x3 3 Cubic 3 Monomial8x4 – 2x3 + 3x 4 Fourth

Degree4 Trinomial

Name that Polynomial!

3x + 4x2

Quadratic binomial

4x – 1 + 5x3 + 7x

Cubic trinomial

2x – 3 – 8x2

Quadratic trinomial

Example 3 and 4

• It is exactly like combining like terms.

• **Remember: 2x2 = 2x3 and they can not be combined.**

Problem 5

(x3 – 3x + 5x) - (7x3 + 5x2 – 12)

One More…

(2g4 – 3g + 9) + (-g3 + 12g)

Multiplying and FactoringLesson 8-2

2x(3x + 1) = 6x2 + 2x

x x x 1 x

x

x2

x2

x2

x2

x2

x2

x

x

What is –x3(9x4 – 2x3 + 7)?

-x3(9x4) + (-x3)(-2x3) + (-x3)(7)

-9x7 + 2x6 + -7x3

Got it?

What is the simpler form of 5n(3n3 – n2 + 8)?

5n(3n3) – 5n(n2) + 5n(8)

15n4 – 5n3 + 40n

Factoring

• Factoring is the opposite of multiplying.

• It “undoes” what we did in the last slide.

What is the GCF of 5x3 + 25x2 + 45x?

5x3 = 5 x x x

25x2 = 5 5 x x

45x = 3 3 5 x

What do they have in common? 5x

Got it?What is the GCF of 3x4 – 9x2 – 12x?

3x4 = 3 x x x x

-9x2 = -1 3 3 x x

-12x = -1 3 4 x

What do they have in common? 3x

Factor 4x5 – 24x3 + 8x

Common(leftover + leftover + leftover)4x5 = 2 2 x x x x x∙ ∙ ∙ ∙ ∙ ∙-24x3 = -1 2 2 2 3 x x x∙ ∙ ∙ ∙ ∙ ∙ ∙8x = 2 2 2 x∙ ∙ ∙

Common: 2 2 x∙ ∙ Left over: x4 – 6x2 + 2

Final Answer: 4x(x4 – 6x2 + 2)

Got it? Factor 9x6 + 15x4 + 12x2

9x6 = 3 3 x x x x x x15x4 = 3 5 x x x x12x2 = 2 2 3 x x

Common: 3x2

Leftover: 3x4 + 5x2 + 4

Final Answer: 3x2(3x4 + 5x2 + 4)

Example 4

Got it?

Multiplying BinomialsLesson 8-3

Multiply: (2x + 1)(x + 2)

x x 1 x

2

2x2 + 5x + 2

Multiply: (2x + 4)(3x - 7)

• Use the distributive property:

2x(3x - 7) + 4(3x - 7)6x2 - 14x + 12x - 28

6x2 - 2x - 28

Got it? Multiply: (x – 6)(4x + 3)

• Use the distributive property:

x(4x + 3) + -6(4x + 3)4x2 + 3x + -24x + -18

4x2 – 21x + -18

Multiply: (x - 3)(4x - 5)

•Make a table:

x -3

4x 4x2 -12x

-5 -5x 15

4x2 – 12x – 5x + 154x2 – 17x + 15

Multiply: (3x + 1)(x + 4)

•Make a table:

3x 1

x 3x2 -x

4 12x 4

3x2 –x – 12x + 43x2 – 13x + 4

Multiply: (5x – 3)(2x + 1)

• Use FOIL (First, Outside, Inside, Last)

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

10x2 + 5x – 6x – 310x2 – x – 3

Got it? Multiply: (3x - 4)(x + 2)

• Use FOIL (First, Outside, Inside, Last)

(3x – 4)(x + 2)(3x)(x) + (3x)(2) + (-4)(x) + (-4)(2)

3x2 + 6x – 4x – 83x2 + 2x – 8

ApplicationA cylinder has the dimensions shown in the diagram. Which polynomial in standard form best describes the total surface area of a cylinder with a radius of (x + 1) and a height of (x + 4)? The formula for a cylinder is 2πr2 + 2πrh, where r is the radius and h is the height.

Application2πr2 + 2πrh

2π(x + 1)2 + 2π(x + 1)(x + 4)2π(x + 1)(x + 1) + 2π (x + 1)(x + 4)2π(x2 + 2x + 1) + 2π(x2 + 5x + 4)

2π (x2 + 2x + 1 + x2 + 5x + 4)2π (2x2 + 7x + 5)

4πx2 + 14πx + 10π

Let’s do number 29 or 30 together….

Multiplying Special CasesLesson 8-4

The Square of a Binomial

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

(x + 6)2 = x2 + 2(x)(6) + 62

= x2 + 12x + 36

(a – b)2 = a2 – 2ab + b2

(x - 5)2 = x2 – 2(x)(5) + 52

= x2 – 10x + 25

Multiply: (x + 3)2

x 3 x

3

x2 + 3x + 3x + 9x2 + 6x + 9

Problem 1a: Multiply: (x + 8)2

(x + 8) = x2 + 2(x)(8) + 82

= x2 + 16x + 64

Using FOIL:

(x + 8)2 = (x + 8)(x + 8)x2 + 8x + 8x + 64

x2 + 16x + 64

Got it? Multiply: (x + 12)2

x2 + 24x+ 144

Problem 1b: Multiply: (x - 7)2

(x - 7) = x2 - 2(x)(7) + 72

= x2 - 14x + 49

Using FOIL:

(x - 7)2 = (x - 7)(x - 7)x2 - 7x - 7x + 49

x2 - 14x + 49

Got it? Multiply: (2x - 9)2

(2x)2 – 2(2x)(9) + 92

4x2 - 36x+ 81

Problem 2: Applying Squares of BinomialsA square patio is surrounded by the brick walkway shown. What is the area of the walkway?

Problem 2: Applying Squares of Binomials (Continued)

Total Area: (x + 6)2 = x2 + 2(x)(6) + 62

Total Area = x2 + 12x + 36

Area of patio: x x = x∙ 2

Area of walkway = Total Area – Patio= x2 + 12x + 36 – x2

= 12x + 36

Problem 3: Using Mental Math

What is 392? Use mental math.

392 = (40 – 1)2(40 – 1)2 = 402 – 2(40(1) + 12

= 1600 – 80 – 1= 1521

Got it?

Use mental math to compute 852.

7225

What if…. (a + b)(a – b)?

(a + b)(a – b) = a2 – b2

(x + 2)(x – 2) = x2 – 4

Using FOIL:(x + 2)(x – 2)

x2 - 2x + 2x + -4x2 – 2

Problem 4:(x + 5)(x – 5)

= x2 – 52

= x2 – 25

(x3 + 8)(x3 – 8)(x3)2 – 82

x6 - 64

Got it?

(x + 9)(x – 9)x2 - 81

(3c – 4)(3c + 4)9c2 - 16

Problem 5

What is 64 x 56 using mental math?

(60 + 4)(60 – 4)602 – 42

3600 – 163584

Factoring x2 + bx + cLesson 8-5

Key Concept: (x + 3)(x + 7)

(x + 3)(x + 7)(x)(x) + 3x + 7x + (3)(7)

x2 + (3 + 7)x + 21

10 is 3 + 7 and 21 is 3 7

Problem 1: Factor x2 + 8x + 15

1. Ask: what are the addition factors of 8? (whole numbers only)

2. What are the multiplication factors of 15? 3. When are they the same?

4. 8 = (1 + 7) (2 + 6) (3 + 5) (4 + 4)5. 15 = (1 x 15) (3 x 5)6. 3 and 5 appear in both. (x + 3)(x + 5)

Got it? Factor r2 + 11r + 24

(r + 3)(r + 8)

Problem 2: Factor x2 - 11x + 24

-11 = (-1 + -10) (-2 + -9) (-3 + -8) (-4 + -7) (-5 + -6)24 = (-2 x -12) (-3 x -8) (-4 x -6)

-3 and -8 appear in both.

Put down parentheses (x )(x ) Fill in what you know (x + -3)(x + -8)Simplify (x – 3)(x – 8)

Got it? Factor y2 - 6y + 8

(y – 4)(y – 2)

Problem 3: Factor x2 + 2x – 15

-15 = (-3 x 5) (3 x -5)2 = (-3 + 5)

Put down parentheses (x )(x ) Fill in what you know (x + -3)(x + 5)Simplify (x – 3)(x + 5)

Got it?

a. n2 + 9n – 36 (n + 12)(n – 3)

b. c2 – 4c – 21

(c – 7)(c + 3)

Problem 4: x2 - 2x – 35

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

Put down parentheses (x )(x ) Fill in what you know (x + -7)(x + 5)Simplify (x – 7)(x + 5)

Got it?

What is the dimensions of a rectangle with an area of x2 – x – 72?

(x + 8)(x – 9)

Problem 5: Factor x2 + 6xy – 55y2

6 = (-5 + 11)-55 = (-5 x 11) (5 x -11)

Put down parentheses (x )(x ) Fill in what you know (x + -5)(x +11)Simplify (x + -5y)(x + 11y)

Got it? Factor m2 + 6mn – 27n2

(m + 9n)(m – 3n)

Factoring ax2 + bx + c

Lesson 8-6

Problem 1Factoring 5x2 + 11x + 2• Guess and CheckWhat numbers multiplied together equal 2?

2 x 1I know I have to have a 2 and 1.

( + 2)( + 1)What numbers multiplied together equal 5?

5 x 1

Problem 1Factoring 5x2 + 11x + 2Try out all the possibilities…

Does (5x + 2)(1x + 1) = 5x2 + 11 + 2?

NO, it equals 5x2 + 7x + 2

Does (5x + 1)(1x + 2) = 5x2 + 11 + 2?

YES

Got it? 1

Try: 6x2 + 13x + 5

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

Problem 2: Factor 3x2 + 4x - 15What factors make up 3?

3 x 1What factors make up -15?

-1 x 15, -3 x 5, -5 x 3, -15 x 1Try all the possibilities:

(3x + -1)(x + 15)(3x + 15)(x + -1)(3x + -3)(x + 5)(3x + 5)(x + -3)

Problem 2: Factor 3x2 + 4x - 15What factors make up 3?

3 x 1What factors make up -15?

-1 x 15, -3 x 5, -5 x 3, -15 x 1Try all the possibilities:

(3x + -5)(x + 3)(3x + 3)(x + -5)

Which one worked? (3x – 5)(x + 3)

Got it? 2

Factor: 10x2 + 31x – 14

Answer: (2x + 7)(5x – 2)

What factors make up 2? 2 x 1

What factors make up -7?-1 x 7, 1 x -7

Try all the possibilities:(2x + -1)(x + 7)(2x + 7)(x + -1)(2x + -7)(x + 1)(2x + 1)(x + -7)

Which one worked? (2x + 1)(x – 7)

Problem 3: Factor 2x2 – 13x – 7

Got it? 3The area of a rectangle is 8x2 + 22x + 15. What are the dimensions. (Factor this trinomial)

Answer: (2x + 3)(4x + 5)

Factor out what they all have in common. 18x2 = 2, 3, 3, x, x-33x = -1, 3, 11, x12 = 2, 2, 3

3(6x2 + -11x + 4)

Problem 4:Factor 18x2 – 33x + 12

3(6x2 + -11x + 4)

Factors of 6: 1 x 6, 2 x 3Factors of 4: -1 x -4, -2 x -2

Guess and Check:(6x – 1)(x – 4)(6x – 4)(x – 1)(6x – 2)(x – 2)

3(6x2 + -11x + 4)

Factors of 6: 1 x 6, 2 x 3Factors of 4: -1 x -4, -2 x -2

Guess and Check:(2x – 1)(3x – 4)

3(2x-1)(3x – 4)

Got it? 8x2 – 36x – 20

Answer: 4(2x + 1)(x – 5)

Factoring Special CasesLesson 8-7

Do you remember that…?

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

a2 - 2ab + b2 = (a - b)(a - b) = (a - b)2

These are Perfect-Square Trinomials.Ex.

x2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)2

Key Concept: Perfect-Square Trinomials:- The first and last terms are perfect squares

4x2 – 12x + 94 and 9 are perfect squares

- The middle term is twice the product of one factor of the first term and one factor of the last term.

Factors of 4 = 2 ∙ 2Factors of 9 = 3 ∙ 3

Middle Term = 2(2)(3) = 12

Problem 1: Factor x2 – 12x + 36This is one way to solve this problem:

Rewrite the trinomial: x2 – 12x + 62

x2 – 2(x)(6) + 62

Does the middle equal -2ab?Write the square of the binomial:

(x – 6)2

Problem 1: Factor x2 – 12x + 36This is another way to solve this problem:

Ask: what factors of 36 make 12?x2 – 12x + 36(x – 6)(x – 6)

(x – 6)2

Got it?

a. x2 + 6x + 9

(x + 3)2

b. x2 – 14x + 49(x – 7)2

Problem 2: Factor 4x2 + 20x + 25

Now, the coefficient of x is 4, so we have to break apart 4 and 25. 4 = 2 x 225 = 5 x 5

(2x)2 + 2(2)(5) + 52

(2x + 5)(2x + 5)(2x + 5)2

Got it?

You are building a square patio. The area of the patio is 16m2 – 72m + 81. What is the length of one side of the patio?

4m - 9

Just like before…

(a2 – b2) = (a + b)(a – b)We learned about this in 8-2.

x2 – 64 = (x + 8)(x – 8)

x2 – 144 = (x + 12)(x – 12)

Problem 3: Factor z2 – 9?

Rewrite 9 as a square: z2 – 32

(z + 3)(z – 3)

Got it?

What is the factored form of v2 – 100?

(v – 10)(v + 10)

What is the factored form of s2 – 16?

(s – 4)(s + 4)

Problem 4: Factor 16x2 – 81

Write each term as a square.(4x)2 – 92

(4x – 9)(4x + 9)

We can always use FOIL to check your answer.

Got it?

Factor 25d2 – 64.

(5d – 8)(5d + 8)

Problem 4: Factor 24g2 – 6

At first, we do not see any squares.Let’s factor: what do they both have in common?

6(4g2 – 1)Take the square of the binomial.

6((2g)2 – 12)6(2g2 + 1)(2g2 – 1)

Got it?

a. 12t2 – 48

12(t + 2)(t – 2)

b. 12x2 + 12x + 3

3(2x + 1)2

Factoring by GroupingLesson 8-8

New Homework Assignment

10 – 27 all…no more, no less

Factoring by Grouping

Your goal is to factor each polynomial so that you have a common binomial.

It’s best to guess and check what parts to group.

Problem 1: Factor by Grouping

3n3 – 12n2 + 2n – 8 Let’s start and group the terms with the

highest degree.

(3n3 – 12n2)+ (2n – 8)Now factor each binomial, using the GCF.

Problem 1: Factor by Grouping

(3n3 – 12n2)+ (2n – 8)

(3n3 – 12n2) = 3n2(n – 4) (2n – 8) = 2(n – 4)

So, 3n2(n – 4) + 2(n – 4) is the result of factoring.

This equals (3n2 + 2)(n – 4).

Problem 1: FYI

If combining these terms did not result in a common binomial, then we would have to

group them up in a different way.

Got it?

8t3 + 14t2 + 20t + 35

(2t2 + 5)(4t + 7)

Problem 2: Factoring Completely

Factor: 4q4 – 8q3 + 12q2 – 24qThey all have a 4q in common.

4q(q3 – 2q2 + 3q – 6)

Now factor the terms that are in ().q3 – 2q2 + 3q – 6

Try grouping the two terms with the highest degree together.

Problem 2: Factoring Completely

(q3 – 2q2)+ (3q – 6)

q2(q – 2) + 3(q – 2)Yea! We grouped the right terms together!

Final Answer:4q(q2 + 3)(q – 2)

Got it?

Factor completely:

6h4 + 9h3 + 12h2 + 18h

3h(h2 + 2)(2h + 3)