Algebra unit 8

UNIT 8 POLYNOMIALSUNIT 8 POLYNOMIALS

Warm UpEvaluate 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

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

Objectives

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

Vocabulary

binomialtrinomial

quadraticcubic

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.

Example 1: Finding the Degree of a Monomial

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.

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

Remember!

Check It Out! Example 1

Find the degree of each monomial.

a. 1.5k2m

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

b. 4x


b. 2c3


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.

Find the degree of each polynomial.

Example 2: Finding the Degree of a 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.



:degree 3 :degree 4

–5: degree 0



a. 5x – 65x: 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



–6: degree 0

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

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.

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

Example 3A: Writing Polynomials in Standard Form

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 3B: Writing Polynomials in Standard Form


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.

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

Remember!

y5 = 1y5

Check It Out! Example 3a


16 – 4x2 + x5 + 9x3


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.

Check It Out! Example 3b



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.

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

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

Example 4: Classifying Polynomials

A. 5n3 + 4nDegree 3 Terms 2

5n3 + 4n is a cubic binomial.

B. 4y6 – 5y3 + 2y – 9Degree 6 Terms 4

4y6 – 5y3 + 2y – 9 is a

6th-degree polynomial.

C. –2xDegree 1 Terms 1

–2x is a linear monomial.



a. x3 + x2 – x + 2Degree 3 Terms 4

x3 + x2 – x + 2 is a cubic polymial.

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

c. –3y8 + 18y5 + 14yDegree 8 Terms 3

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

A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds?

Example 5: Application

Substitute the time for t to find the lip balm’s height. –16t2 + 220

–16(3)2 + 220 The time is 3 seconds.

–16(9) + 220Evaluate the polynomial by using

the order of operations.–144 + 22076

A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds?

Example 5: Application Continued

After 3 seconds the lip balm will be 76 feet from the water.

Check It Out! Example 5 What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?

Substitute the time t to find the firework’s height.–16t2 + 400t + 6

–16(5)2 + 400(5) + 6 The time is 5 seconds.

–16(25) + 400(5) + 6

–400 + 2000 + 6 Evaluate the polynomial by using the order of operations.

–400 + 20061606

Check It Out! Example 5 Continued

What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?

When the firework explodes, it will be 1606 feet above the ground.

Lesson Quiz: Part I


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

Lesson Quiz: Part II


5. 18x2 – 12x + 5 quadratic trinomial

6. 2x4 – 1 quartic binomial

7. The polynomial 3.675v + 0.096v2 is used to estimate the stopping distance in feet for a car whose speed is y miles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727.65 ft

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Algebra unit 8

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