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Algebra 1 ~ Chapter 8.4

Polynomials

Remember: A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. “Mono” – single term

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

Ex. 1 - Find the degree of each monomial.

A. 4p4q3

The degree is 7.

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

B. 7ed A variable written without an exponent has an exponent of 1. 1+ 1 = 2.

C. 3There is no variable, but you can

write 3 as 3x0.

The degree is 2.

The degree is 0.

* A polynomial is the sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree. “poly” – many

An example of a polynomial is 3a + 4b – 8c

That expression consists of three monomials “combined” with addition or subtraction.

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

Ex. 2 – Find the degree of each polynomials. Then name the polynomials based on # of terms.A.) 5m4 + 3m

B.) -4x3y2 + 3x2 + 5

C.) 3a + 7ab – 2a2b

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

The degree of the polynomial is 5.

The degree of the polynomial is 3.

This polynomial has 2 terms, so it is a binomial.This polynomial has 3 terms, so it is a trinomial.This polynomial has 3 terms, so it is a trinomial.

Writing Polynomials in Order

The terms of a polynomial are usually arranged so that the powers of one variable are in ascending (increasing) order or descending (decreasing) order.

Ex. 3 – Arrange the terms of the polynomial so

that the powers of x are in descending order.

6x – 7x5 + 4x2 + 9

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

6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9

Degree 1 5 2 0 5 2 1 0

The polynomial written in descending order is -7x5 + 4x2 + 6x + 9.

Ex. 4 - Write the terms of the polynomial so that the powers of x are in descending order.

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

y2 + y6 − 3y

y2 + y6 – 3y y6 + y2 – 3y

Degree 2 6 1 6 2 1

The polynomial written in descending order is

y6 + y2 – 3y.

6-2 Adding and Subtracting Polynomials

Algebra 1 ~ Chapter 8.5

“Adding and Subtracting Polynomials”

6-2 Adding and Subtracting Polynomials

Warm Up - Simplify each expression by combining like terms.

1. 4x + 2x

2. 3y + 7y

3. 8p – 5p

4. 5n + 6n2

5. 3x2 + 6x2

6. 12xy – 4xy

6x

10y

3p

Not like terms

9x2

8xy

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

Example 1: Adding and Subtracting Monomials

A. 12p3 + 11p2 + 8p3

12p3 + 8p3 + 11p2

20p3 + 11p2

B. 5x2 – 6 – 3x + 8

5x2 – 3x + 8 – 6

5x2 – 3x + 2

Arrange the terms so the “like” terms are next to each other and then simplify.

Polynomials can be added in either vertical or horizontal form.

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

In vertical form, align the like terms and add:

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

7x2 + 9x + 3

In horizontal form, regroup and combine like terms.

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

+ 2)= 7x2 + 9x + 3

Example 2: Adding Polynomials

A. (4m2 + 5m + 1) + (m2 + 3m + 6)

(4m2 + 5m + 1) + (m2 + 3m + 6)

(4m2 + m2) + (5m + 3m) + (1 + 6)

5m2 + 8m + 7

B. (10xy + x) + (–3xy + y)

(10xy + x) + (–3xy + y)

(10xy – 3xy) + x + y

7xy + x + y

Example 2: Adding Polynomials

C.

Subtracting PolynomialsSimplify (4x + 5) – ( 2x + 1)

(4x – 2x) + (5 – 1 )

2x + 4(4x + 5) + (-2x – 1)

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

2x + 4

Option #1: Option #2: Recall that you can subtract a number by adding its opposite.

Example 3: Subtracting Polynomials

A. (4m2 + 5m + 1) − (m2 + 3m + 6)

(4m2 + 5m + 1) − (m2 + 3m + 6)

(4m2 − m2) + (5m − 3m) + (1 − 6)

3m2 + 2m – 5

B. (10x3 + 5x + 6) − (–3x3 + 4)

(10x3 - - 3x3) + (5x – 0x) + (6 – 4)

13x3 + 5x + 2

Example 3C: Subtracting Polynomials

(7m4 – 2m2) – (5m4 – 5m2 + 8)

(7m4 – 5m4) + (−2m2 – −5m2) + (0 – 8)

(7m4 – 5m4) + (–2m2 + 5m2) – 8

2m4 + 3m2 – 8

Example 3D: Subtracting Polynomials

(–10x2 – 3x + 7) – (x2 – 9)

(–10x2 – x2) + (−3x – 0x) + (7 – -9)

–11x2 – 3x + 16

Lesson Wrap Up

Simplify each expression.

1. 7m2 + 3m + 4m2

2. (r2 + s2) – (5r2 + 4s2)

3. (10pq + 3p) + (2pq – 5p + 6pq)

4. (14d2 – 8) – (6d2 – 2d + 1)

–4r2 – 3s2

11m2 + 3m

18pq – 2p

8d2 +2d – 9

5. (2.5ab + 14b) – (–1.5ab + 4b) 4ab + 10b

Assignment Study Guide 8-4 (In-Class) Study Guide 8-5 (In-Class) Skills Practice 8-4 (Homework) Skills Practice 8-5 (Homework)