Adding and Subtracting Polynomials

Section 8-1

Goals

Goal •  To classify, add, and

subtract polynomials.

Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary •  Monomial •  Degree of a Monomial •  Polynomial •  Standard Form of a Polynomial •  Degree of a Polynomial •  Binomial •  Trinomial

A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.

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

Definitions

Find the degree of each monomial.

A. 4p4q3

The degree is 7. Add the exponents of the variables: 4 + 3 = 7.

B. 7ed

The degree is 2. Add the exponents of the variables: 1+ 1 = 2.

C. 3

The degree is 0. Add the exponents of the variables: 0 = 0.

Example: Degree of a Monomial

Find the degree of each monomial.

a. 1.5k2m

The degree is 3. Add the exponents of the variables: 2 + 1 = 3.

b. 4x

The degree is 1. Add the exponents of the variables: 1 = 1.

b. 2c3

The degree is 3. Add the exponents of the variables: 3 = 3.

Your Turn:

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.

Add or subtract like terms by adding or subtracting the coefficients of the like terms.

Like Terms

Identify the like terms in each polynomial.

A. 5x3 + y2 + 2 – 6y2 + 4x3

B. 3a3b2 + 3a2b3 + 2a3b2 – a3b2

Identify like terms. 5x + y + 2 – 6y + 4x 3 2 2 3

Like terms: 5x3 and 4x3, y2 and –6y2

3a b + 3a b + 2a b – a b 3 2 2 3 3 3 2 2 Identify like terms.

Like terms: 3a3b2, 2a3b2, and –a3b2

Example: Identify Like Terms

Identify the like terms in the polynomial.

C. 7p3q2 + 7p2q3 + 7pq2

Identify like terms.

There are no like terms.

7p3q2 + 7p2q3 + 7pq2

Example: Identify Like Terms

Identify the like terms in each polynomial.

A. 4y4 + y2 + 2 – 8y2 + 2y4

B. 7n4r2 + 3n2r3 + 5n4r2 + n4r2

Identify like terms. 4y + y + 2 – 8y + 2y 4 2 2 4

Like terms: 4y4 and 2y4, y2 and –8y2

7n4r2 + 3n2r3 + 5n4r2 + n4r2 Identify like terms.

Like terms: 7n4r2, 5n4r2, and n4r2

Your Turn:

Identify the like terms in the polynomial.

C. 9m3n2 + 7m2n3 + pq2

Identify like terms.

There are no like terms.

9m3n2 + 7m2n3 + pq2

Your Turn:

Simplify.

A. 4x2 + 2x2

2 6x Combine coefficients:

4 + 2 = 6

Identify like terms. 4x2 + 2x2

Example: Add or Subtract Monomials

Simplify.

B. 3n5m4 - n5m4

Identify like terms. 3n5m4 - n5m4

Combine coefficients: 3 - 1 = 2. 2n5m4

Example: Add or Subtract Monomials

Simplify.

A. 2x3 - 5x3

Identify like terms.

Combine coefficients: 2 - 5 = -3

2x3 - 5x3

-3x3

Your Turn:

Simplify.

B. 2n5p4 + n5p4

Identify like terms.

Combine coefficients: 2 + 1 = 3

2n5p4 + n5p4

3n5p4

Your Turn:

A polynomial is a monomial or a sum or difference of monomials.

Example: 3x4 + 5x2 – 7x + 1 This polynomial is the sum of the monomials 3x4, 5x2, -7x, and 1.

The degree of a polynomial is the degree of the term with the greatest degree.

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

Definitions

Find the degree of each polynomial. A. 11x7 + 3x3

11x7: degree 7 3x3: degree 3 The degree of the polynomial is the greatest degree, 7.

Find the degree of each term.

B.

Find the degree of each term.

The degree of the polynomial is the greatest degree, 4.

:degree 3 :degree 4

–5: degree 0

Example: Degree of a Polynomial

Find the degree of each polynomial. a. 5x – 6

5x: degree 1 Find the degree of

each term. The degree of the polynomial is the greatest degree, 1.

b. x3y2 + x2y3 – x4 + 2 x3y2: degree 5

The degree of the polynomial is the greatest degree, 5.

Find the degree of each term.

–6: degree 0

x2y3: degree 5 –x4: degree 4 2: degree 0

Your Turn:

The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.

The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. 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.

Definitions

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

6x – 7x5 + 4x2 + 9 Find the degree of each term. Then arrange them in descending order:

6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9 Degree 1 5 2 0 5 2 1 0

–7x5 + 4x2 + 6x + 9. The standard form is The leading coefficient is –7.

Example: Standard Form

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

16 – 4x2 + x5 + 9x3 Find the degree of each term. Then arrange them in descending order:

16 – 4x2 + x5 + 9x3 x5 + 9x3 – 4x2 + 16 Degree 0 2 5 3 0 2 3 5

The standard form is The leading coefficient is 1.

x5 + 9x3 – 4x2 + 16.

Your Turn:

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

Find the degree of each term. Then arrange them in descending order:

18y5 – 3y8 + 14y

18y5 – 3y8 + 14y –3y8 + 18y5 + 14y Degree 5 8 1 8 5 1

The standard form is The leading coefficient is –3.

–3y8 + 18y5 + 14y.

Your Turn:

Some polynomials have special names based on their degree and the number of terms they have.

Degree Name 0

1

2

Constant

Linear

Quadratic

3

4

5

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

Cubic

Quartic

Quintic

Name Terms Monomial

Binomial

Trinomial

Polynomial 4 or more

1

2

3

By Degree

By # of Terms

Classify each polynomial according to its degree and number of terms.

A. 5n3 + 4n Degree 3 Terms 2

5n3 + 4n is a cubic binomial.

B. 4y6 – 5y3 + 2y – 9 Degree 6 Terms 4

4y6 – 5y3 + 2y – 9 is a 6th-degree polynomial.

C. –2x Degree 1 Terms 1

–2x is a linear monomial.

Example: Classifying Polynomials

Classify each polynomial according to its degree and number of terms.

a. x3 + x2 – x + 2

Degree 3 Terms 4 x3 + x2 – x + 2 is a cubic polynomial.

b. 6 Degree 0 Terms 1 6 is a constant monomial.

c. –3y8 + 18y5 + 14y Degree 8 Terms 3

–3y8 + 18y5 + 14y is an 8th-degree trinomial.

Your Turn:

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

Combine like terms.

A. 12p3 + 11p2 + 8p3 12p3 + 11p2 + 8p3 12p3 + 8p3 + 11p2 20p3 + 11p2

Identify like terms. Rearrange terms so that like

terms are together. Combine like terms.

B. 5x2 – 6 – 3x + 8 5x2 – 6 – 3x + 8 5x2 – 3x + 8 – 6 5x2 – 3x + 2

Identify like terms. Rearrange terms so that like

terms are together. Combine like terms.

Example: Simplifying Polynomials

Combine like terms.

C. t2 + 2s2 – 4t2 – s2

t2 – 4t2 + 2s2 – s2 t2 + 2s2 – 4t2 – s2

–3t2 + s2

Identify like terms. Rearrange terms so that like

terms are together. Combine like terms.

D. 10m2n + 4m2n – 8m2n

10m2n + 4m2n – 8m2n

6m2n

Identify like terms.

Combine like terms.

Example: Simplifying Polynomials

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

Remember!

a. 2s2 + 3s2 + s

Combine like terms.

2s2 + 3s2 + s 5s2 + s

b. 4z4 – 8 + 16z4 + 2 4z4 – 8 + 16z4 + 2

4z4 + 16z4 – 8 + 2 20z4 – 6

Identify like terms.

Combine like terms.

Identify like terms. Rearrange terms so that like

terms are together. Combine like terms.

Your Turn:

c. 2x8 + 7y8 – x8 – y8

Combine like terms.

2x8 + 7y8 – x8 – y8 2x8 – x8 + 7y8 – y8 x8 + 6y8

d. 9b3c2 + 5b3c2 – 13b3c2

9b3c2 + 5b3c2 – 13b3c2 b3c2

Identify like terms.

Combine like terms.

Identify like terms. Rearrange terms so that like

terms are together. Combine like terms.

Your Turn:

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

= 7x2 + 9x + 3

5x2 + 4x + 1 + 2x2 + 5x + 2

7x2 + 9x + 3

Adding Polynomials

Add.

A. (4m2 + 5) + (m2 – m + 6)

(4m2 + 5) + (m2 – m + 6)

(4m2 + m2) + (–m) +(5 + 6)

5m2 – m + 11

Identify like terms. Group like terms

together. Combine like terms.

B. (10xy + x) + (–3xy + y)

(10xy + x) + (–3xy + y)

(10xy – 3xy) + x + y 7xy + x + y

Identify like terms. Group like terms

together. Combine like terms.

Example: Adding Polynomials

Add.

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

Identify like terms.

Group like terms together within each polynomial.

Combine like terms.

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

(6x2 + 3x2 – 8x2) + (3y – 4y – 2y)

Use the vertical method. 6x2 – 4y + –5x2 + y

x2 – 3y Simplify.

Example: Adding Polynomials

Add.

Identify like terms.

Group like terms together.

Combine like terms.

Your Turn:

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

(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)

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

12a3 + 15a2 – 16a

Identify like terms.

Group like terms together.

Combine like terms.

Your Turn:

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

Subtract.

(x3 + 4y) – (2x3)

(x3 + 4y) + (–2x3)

(x3 + 4y) + (–2x3)

(x3 – 2x3) + 4y

–x3 + 4y

Rewrite subtraction as addition of the opposite.

Identify like terms.

Group like terms together.

Combine like terms.

Example: Subtracting Polynomials

Subtract.

(7m4 – 2m2) – (5m4 – 5m2 + 8)

(7m4 – 2m2) + (–5m4 + 5m2 – 8)

(7m4 – 5m4) + (–2m2 + 5m2) – 8

(7m4 – 2m2) + (–5m4 + 5m2 – 8)

2m4 + 3m2 – 8

Rewrite subtraction as addition of the opposite.

Identify like terms.

Group like terms together.

Combine like terms.

Example: Subtracting Polynomials

Subtract. (–10x2 – 3x + 7) – (x2 – 9)

(–10x2 – 3x + 7) + (–x2 + 9)

(–10x2 – 3x + 7) + (–x2 + 9)

–10x2 – 3x + 7 –x2 + 0x + 9

–11x2 – 3x + 16

Rewrite subtraction as addition of the opposite.

Identify like terms.

Use the vertical method. Write 0x as a placeholder. Combine like terms.

Example: Subtracting Polynomials

Subtract. (9q2 – 3q) – (q2 – 5)

(9q2 – 3q) + (–q2 + 5)

(9q2 – 3q) + (–q2 + 5)

9q2 – 3q + 0 + − q2 – 0q + 5

8q2 – 3q + 5

Rewrite subtraction as addition of the opposite.

Identify like terms. Use the vertical method. Write 0 and 0q as

placeholders.

Combine like terms.

Your Turn:

Subtract.

(2x2 – 3x2 + 1) – (x2 + x + 1)

(2x2 – 3x2 + 1) + (–x2 – x – 1)

(2x2 – 3x2 + 1) + (–x2 – x – 1)

–x2 + 0x + 1 + –x2 – x – 1

–2x2 – x

Rewrite subtraction as addition of the opposite.

Identify like terms.

Use the vertical method. Write 0x as a placeholder.

Combine like terms.

Your Turn:

Joke Time

•  How does Hitler tie his shoes? •  With little Nazis!

•  What did one snowman say to the other? •  Do you smell carrots?

•  There’s two fish in a tank. What did one fish say to the other?

•  You man the guns, I’ll drive!

Assignment

•  8-1 Exercises Pg. 488 - 489: #8 – 44 even