7-6 Polynomials

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Lesson Presentation

California Standards

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7-6 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; x = 3

Identify the coefficient in each term.

5. 4x3 6. y3

7. 2n7 8. –s4

7 13

2 69

4 1

2 –1

7-6 Polynomials

California Standards

Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques.

7-6 Polynomials

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

Vocabulary

binomialtrinomial

quadraticcubic

roots

7-6 Polynomials

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

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

7-6 Polynomials

Additional 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.A variable written without an

exponent has an exponent of 1. 1+ 1 = 2.

C. 3

The degree is 0.There is no variable, but you

can write 3 as 3x0.

7-6 Polynomials

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

Remember!

7-6 PolynomialsCheck 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

The degree is 1.Add the exponents of the

variables: 1 = 1.

c. 2c3

The degree is 3.Add the exponents of the

variables: 3 = 3.

7-6 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.

7-6 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.

7-6 Polynomials

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

Additional Example 2A: 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.

7-6 Polynomials

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

Additional Example 2B: Writing Polynomials in Standard Form

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.

7-6 Polynomials

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

Remember!

y5 = 1y5

7-6 PolynomialsCheck It Out! Example 2a

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 0235

The standard form is The leading coefficient is 1.

x5 + 9x3 – 4x2 + 16.

7-6 PolynomialsCheck It Out! Example 2b

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.

7-6 Polynomials

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

7-6 Polynomials

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

Additional Example 3: Classifying Polynomials

A. 5n3 + 4nDegree 3 Terms 2

5n3 + 4n is a cubic binomial.

B. –2xDegree 1 Terms 1

–2x is a linear monomial.

7-6 Polynomials

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

Check It Out! Example 3

a. x3 + x2 – x + 2Degree 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 + 14yDegree 8 Terms 3

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

7-6 Polynomials

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?

Additional Example 4: Application

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

–16(3)2 + 200 The time is 3 seconds.

–16(9) + 200Evaluate the polynomial by using

the order of operations.–144 + 20076

7-6 Polynomials

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?

Additional Example 5 Continued

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

7-6 PolynomialsCheck It Out! Example 4

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 for 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

–400 + 20061606

7-6 PolynomialsCheck It Out! Example 4 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.

7-6 Polynomials

A root of a polynomial in one variable is a value of the variable for which the polynomial is equal to 0.

7-6 Polynomials

Additional Example 5: Identifying Roots of Polynomials

Tell whether each number is a root of 3x2 – 48.

A. 4

3x2 – 48

3(4)2 – 48

3(16) – 48

48 – 48

4 is a root of 3x2 – 48.

0

Substitute for x.

Simplify.

B. 0

3x2 – 48

3(0)2 – 48

3(0) – 48

0 – 48

–48 0 is not a root of 3x2 – 48.

7-6 Polynomials

Additional Example 5: Identifying Roots of Polynomials

Tell whether each number is a root of 3x2 – 48.

C. –4

3x2 – 48

3(–4)2 – 48

3(16) – 48

48 – 48

–4 is a root of 3x2 – 48. 0

Substitute for x.

Simplify.

7-6 PolynomialsCheck It Out! Example 5

Tell whether 1 is a root of 3x3 + x – 4.

3x3 + x – 4

3(1)3 + (1) – 4

3(1) + 1 – 4

3 + 1 – 4

1 is a root of 3x3 + x – 4. 0

Substitute for x.

Simplify.

7-6 PolynomialsLesson Quiz: 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

7-6 PolynomialsLesson Quiz: Part II

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

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 v miles per hour on flat, dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727.65 ft

7-6 PolynomialsLesson Quiz: Part IIl

Tell whether each number is a root of 3p2 – 8 + 4.

8. 2

9. –2

yes

no