Topic 8: Polynomials

Table of Contents1. Introduction to Polynomials

2. Adding & Subtracting Polynomials

3. Multiplying Polynomials

4. Special Products of Binomials

5. Factoring Polynomials

6. Factoring Polynomials, part 2

7. Factoring by Grouping

Introduction to Polynomials

Monomial: a number, a variable, or a product of numbers and variables with whole-number exponents.

Degree of a monomial: is the sum of the exponents of the variables. A constant has degree 0.

Vocab

Polynomial: an expression of more than two algebraic terms.Example: 3x4 + 5x2 – 7x + 1

Degree of a polynomial is the degree of the term with the greatest power/exponent.

Example: The degree of 3x4 + 5x2 – 7x + 1 is 4.

Vocab

Find the degree of each polynomial.

1. 11x7 + 3x3

2.

Degree of a Polynomial

Standard form of a polynomial: Polynomial written with the terms in order from greatest degree to least degree.

Leading Coefficient: When written in standard form, the coefficient of the first term is called the leading coefficient.

Example: 3x4 + 5x2 – 7x + 1 and 3 is the leading coefficient.

Vocab

Write the polynomial in standard form. Then give the leading coefficient.

3. 6x – 7x5 + 4x2 + 9

4. 16 – 4x2 + x5 + 9x3

5. 18y5 – 3y8 + 14y

Let’s Practice…

Special Polynomial Names

Degree Name

0

1

2

Constant

Linear

Quadratic

3

4

5

6 or more 6th,7th,degree and so on

Cubic

Quartic

Quintic

NameTerms

Monomial

Binomial

Trinomial

Polynomial4 or more

1

2

3

By Degree

By # of Terms

You can add or subtract monomials by adding or subtracting like terms.

4a3b2 + 3a2b3 – 2a3b2

Like terms

Not like terms

The variables have the same powers.

The variables have different powers.

Review: Like Terms

Identify the like terms in each polynomial.

6. 5x3 + y2 + 2 – 6y2 + 4x3

7. 3a3b2 + 3a2b3 + 2a3b2 – a3b2

Like terms: ______________________

Identify Like Terms

Like terms: _______________________

8.

Adding and Subtracting Polynomials

Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

Adding and Subtracting Polynomials

Like terms are constants or terms with the same variable(s) raised to the same power(s).

Remember!

Combine like terms.

1. 12p3 + 11p2 + 8p3

2. 5x2 – 6 – 3x + 8

Simplifying Polynomials

3. 2s2 + 3s2 + s – 3s2 – 5s

Combine like terms.

4. 4z4 – 8 + -2z2 +16z4 + 2 + 5z3 – 7

Let’s Practice…

Polynomials can be added in either vertical or horizontal form.

In vertical form, align the like terms and add:

In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms.

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

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

= 7x2 + 9x + 3

5x2 + 4x + 1

+ 2x2 + 5x + 27x2 + 9x + 3

2 Methods: Adding Polynomials

Add.

5. (4m2 + 5) + (m2 – m + 6)

6. (10xy + x) + (–3xy + y)

Adding Polynomials

Add.

Let’s Practice…

8. (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)

7.

To subtract polynomials, remember that subtracting is the same as adding the opposite (distributing the negative). To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:

–(2x3 – 3x + 7)= –2x3 + 3x – 7

Subtracting Polynomials

9. (–10x2 – 3x + 7) – (x2 – 9)

Subtracting Polynomials

10. (x3 + 4y) – (2x3)

Subtracting Polynomials

11. (7m4 – 2m2) – (5m4 – 5m2 + 8)

13. (2x2 – 3x2 + 1) – (x2 + x + 1)

Let’s Practice…

12. (9q2 – 3q) – (q2 – 5)

14.

15. The revenue made by Ford Motor Company from the sale of y cars is given by 2y2 + 8y. The cost to produce y cars is given by the polynomial y2 + 8y -1200. Write a polynomial expression for the profit from making and selling y cars.

16.

17.

Multiplying Polynomials

F.O.I.L

Multiplying Polynomials

Each term in the first polynomial, must be multiplied by each term in the second polynomial.

Method 1: Distribute

First

Outer

Inner

Last

• Multiply!!!

“F.O.I.L.”

Method 2: Box

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

1. Draw a box.

2. Write a polynomial on the top and side of a box.

3. Multiply.

4. Combine like terms.

3x +4

5x

-3

Pick Your Method: (7p - 2)(3p - 4)

Pick Your Method: (7p - 2)(3p - 4)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p

-6p

+8

Let’s Practice…

1. (7x – 10)(3x + 8)

2. (2x – 3)(4x - 8)

3. (5x - 10)(2x + 8)

Multiplying Terms with Exponents

• When FOILing, add the exponents and multiply coefficients.

• Add the little numbers and multiply the big numbers!!!

Example:

(3x2 + 10x)(5x3 – 7x2)

15x5 - 21x4 + 50x4 – 70x3

15x5 + 29x4 – 70x3

Let’s Practice…

4. (7x2 – 10x)(3x3 + 8x2)

5. (2x4 – 3x2)(4x - 8)

6. (5x3 + 2x2)(8x - 7)

Multiplying Larger Polynomials

Each term in the 1st polynomial must be multiplied by each term in the 2nd.

Example:

(7x2 + 2x + 8)(4x3 – 9x2)

Method 2:

Multiply: (2x - 5)(x2 - 5x + 4)

Let’s Practice…

7. (5x2 + 7) (2x3 – 5x2 +9)

Let’s Practice…

8. (10x4 – 5x2 + 8) (8x3 -3x -6)

11. Cameron is building a garden. He designs a rectangular garden with a length of (x + 6) feet and a width of (x +2) feet. When x = 5, what is the area of the garden?

12. Sarah manages a manufacturing plant. From 1990 through 2005, the number of units produced (in thousands) can be modeled by N(x) = 3x2 + 2x, where x is the number of years since 1990. The average cost per unit in dollars can be modeled by C(x) = -x2 + x – 5, where x is the number of years since 1990. Write a polynomial that can be used to model Sarah’s total manufacturing cost for those years.

Special Products of Binomials

Multiply:

1. (2x + 3)(2x + 3)

2. (3x – 4)(3x + 4)

3. (x + 5)(x + 5)

Multiply:

4. (6x – 1)(6x – 1)

5. (3x + 2y)2

Special Products of BinomialsName Algebraically Words

Positive Perfect Square (a + b)2

Negative Perfect Square(a - b)2

Difference of 2 Squares(a + b)(a – b)

6.

Let’s Practice….7. (x + 8)2

8. (4x + 6y)2

9. (-x + 5)2

10. (7x - 3)(7x + 3)

Let’s Practice #211. (x - 3)2

12. (2x + 4y)2

13. (x + 5)(x – 5)

14. (2x + 4)2

15. A square swimming pool is surrounded by a cement walkway with a uniform width. The swimming pool has a side length of (x – 2) feet. The side length of the entire square area including the pool and the walkway is (x + 1) feet. Write an expression for the area of the walkway. Then find the area of the cement walkway when x = 7 feet.

16

17.

18.

Factoring Polynomials

Vocab: Factoring

Factoring is rewriting an expression as a product of factors.

It is the reverse of multiplying polynomials FOILing.

2x x cb= + +

To determine the factors, ask yourself…

What two #’s add to the middle number AND multiply to the

last number?!?!

Let’s Practice…

2 3 2x x+ +

What adds (or subtracts) to get 3 and multiplies to

get 2?

2 7 10x x− +

What adds (or subtracts) to get -7 and

multiplies to get 10?

2 7 44x x− −

What adds (or subtracts) to get -7 and multiplies to get

-44?

Let’s Practice…

Factor:

1. x2 + 5x + 6

2. x2 -7x + 10

3. x2 -11x +24

2x cbx+ +

Signs of Factors

b c Factors

+ + +,+

- -+,- (The factor w/ the greater absolute value is -)

+ -+,- (The factor w/ the greater absolute value is +)

- + -, -

Vocab: GCF

The greatest common factor (GCF) is a common factor of the terms in the expression.

Example:

29 9 18x x+ −

24 8 12x x+ +

Vocab: PrimeIf a polynomials is “prime” it means there are no factors.

Factor or identify as prime.

4. x2 + 7x + 9

5. 2x2 + 10x + 8

6. x2 + 9x + 10

Let’s Practice….

Factor.

7. y2 -10y +16

8. r2 -11r +24

Let’s Practice….

Factor.

9. n2 -15n +56

10. 2v2 + 10v -72

Let’s Practice…

11. x2 + 12x + 36

12. x2 - 8x + 16

Let’s Practice…

13. -2x2 +22x -36

14. 16- x2

Factoring Polynomials,

Part 2

(2 1)(3 5)x x+ + 26 10 3 5x x x= + + +

26 13 5x x= + +

Expanded Form

When factoring problems where a ≠ 1, we first want to get the problem into expanded form before we try to factor.

Expanded Form

Creating Expanded Form

Step 1: Multiply a·c

Step 2: To get to expanded form ask yourself “What multiplies to get a·c, and add/subtracts to get to b.”

Example:

1. Expand: 2x2 +9x +7

2. Expand: 3x2 + 2x – 8

Method 1:

Step 3: Write your new factors in place of bx.

Step 4: Group the first two terms together and the last two terms together.

Step 5: Factor each group

Step 6: Factor again to get the complete factorization

Method 1: 6x2 + 13 x +5 1) Multiply a·c (6·5=30)

2) To get to expanded form ask yourself “What multiplies to get a·c, and add/subtracts to get to b.” (10, 3)

3) Write your new factors in place of bx. (6x2+10x+3x+5)

4) Group the first two terms together and the last two terms together. [(6x2+10x)+(3x+5)]

5) Factor each group [2x(3x+5)+1(3x+5)]

6) Factor again to get the complete factorization [(3x+5)(2x+1)]

Method 2: 6x2 + 13 x +5

Step 3: Fill in box.

Step 4: Factor horizontally and vertically.

Step 5: Terms outside of box are the solution.

Original 1st Term Expanded Term 1

Expanded Term 2 Original Last Term

#1 Factor: 2x2 + 5x -12 Original 1st Term Expanded Term 1

Expanded Term 2 Original Last Term

#2 Factor: 3x2 + 7x +2

#3 Factor: 2x2 + 15x -8

#4 Factor: 16x2 + 28x +10

5

Factoring by Grouping

Factoring by Grouping – Using the distributive property to factor polynomials with

four or more terms.

– Terms can be put into groups and then factored---- each group will have a “like” factor used in regrouping.

A polynomial can be factored by grouping if all of the following conditions exist.

1. There are four or more terms.

2. Terms have common factors that can be grouped together, and

3. There are two common factors that are identical.

Symbols: ax + bx + ay + by = (ax + bx) + (ay + by)

= x(a + b) + y(a + b)

= (x + y)(a + b)

Group, factor

Regroup

Factoring by Grouping

1. Factor each polynomial by grouping. Check your answer.

6h4 – 4h3 + 12h – 8

Factor by Grouping

2. Factor each polynomial by grouping.

5y4 – 15y3 + y2 – 3y

Factor by Grouping

3. Factor each polynomial by grouping.

6b3 + 8b2 + 9b + 12

Let’s Practice…

4. Factor each polynomial by grouping.

4r3 + 24r + r2 + 6

Let’s Practice…

5. 2x3 – 12x2 + 18 – 3x

Factoring with Opposite Groups

Factor each polynomial. Check your answer.

6. 15x2 – 10x3 + 8x – 12

Let’s Practice…

Factor each polynomial by grouping.

7. 2x3 + x2 – 6x – 3

8. 7p4 – 2p3 + 63p – 18

Let’s Practice…

Factoring Procedure