Holt Algebra 1

7-5 Polynomials

Warm Up 4-28-08Evaluate each expression for the given value of x.

1. 2x + 3; x = 2 2. x2 + 4; x = –3

3. –4x – 2; x = –1 4. 7x2 + 2x = 3

Identify the coefficient in each term.

5. 4x3 6. y3

7. 2n7 8. –54

7 13

2 69

4 1

2 –1

Holt Algebra 1

7-5 Polynomials

Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions.

Objectives

Holt Algebra 1

7-5 Polynomials

monomialdegree of a monomialpolynomialdegree of a polynomialstandard form of a polynomialleading coefficient

Vocabulary

binomialtrinomial

quadraticcubic

Holt Algebra 1

7-5 Polynomials

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.

Holt Algebra 1

7-5 Polynomials

Directions: Find the degree of each monomial.

Holt Algebra 1

7-5 Polynomials

Example 1

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.

Holt Algebra 1

7-5 Polynomials

The terms of an expression are the parts being added or subtracted. See Lesson 1-7.

Remember!

Holt Algebra 1

7-5 Polynomials

Example 2

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.

Holt Algebra 1

7-5 Polynomials

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

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

Holt Algebra 1

7-5 Polynomials

Directions: Find the degree of each polynomial.

Holt Algebra 1

7-5 Polynomials

Example 3

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

Holt Algebra 1

7-5 Polynomials

Example 4

a. 5x – 6

5x: degree 1Find 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

Holt Algebra 1

7-5 Polynomials

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.

Holt Algebra 1

7-5 Polynomials

Directions:

Step 1: Find the degree of EACH term

Step 2: Write in standard form by arranging terms in descending order

Step 3: Determine the LEADING coefficient.

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

Holt Algebra 1

7-5 Polynomials

Example 5

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.

Holt Algebra 1

7-5 Polynomials

Example 6

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

y2 + y6 − 3y

y2 + y6 – 3y y6 + y2 – 3y

Degree 2 6 1 2 16

The standard form is The leading coefficient is 1.

y6 + y2 – 3y.

Holt Algebra 1

7-5 Polynomials

A variable written without a coefficient has a coefficient of 1.

Remember!

y5 = 1y5

Holt Algebra 1

7-5 Polynomials

Example 7

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 0235

The standard form is The leading coefficient is 1.

x5 + 9x3 – 4x2 + 16.

Holt Algebra 1

7-5 Polynomials

Example 8

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.

Holt Algebra 1

7-5 Polynomials

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

NameTerms

Monomial

Binomial

Trinomial

Polynomial4 or more

1

2

3

Holt Algebra 1

7-5 Polynomials

Directions:

Step 1: Determine degree of the polynomial or monomial (term with the greatest degree) – first name

Step 2: Count how many total terms there are – last name

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

Holt Algebra 1

7-5 Polynomials

Example 9

A. 5n3 + 4nDegree 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. –2xDegree 1 Terms 1

–2x is a linear monomial.

Holt Algebra 1

7-5 Polynomials

Example 10

a. x3 + x2 – x + 2Degree 3 Terms 4

x3 + x2 – x + 2 is a cubic polymial.

b. 6

Degree 0 Terms 1 6 is a constant monomial.

c. –3y8 + 18y5 + 14yDegree 8 Terms 3

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

Holt Algebra 1

7-5 Polynomials

Lesson Summary: Part I

Find the degree of each polynomial.

1. 7a3b2 – 2a4 + 4b – 15

2. 25x2 – 3x4

Write each polynomial in standard form. Then

give the leading coefficient.

3. 24g3 + 10 + 7g5 – g2

4. 14 – x4 + 3x2

4

5

–x4 + 3x2 + 14; –1

7g5 + 24g3 – g2 + 10; 7

Holt Algebra 1

7-5 Polynomials

Lesson Summary: Part II

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

5. 18x2 – 12x + 5 quadratic trinomial

6. 2x4 – 1 quartic binomial