Holt McDougal Algebra 1 6-3 Polynomials 6-3 Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

Holt McDougal Algebra 1

6-3 Polynomials6-3 Polynomials

Holt Algebra 1

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz



6-3 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 –5


6-3 Polynomials

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

Objectives


6-3 Polynomials

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

Vocabulary

binomialtrinomial

quadraticcubic


6-3 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.


6-3 Polynomials

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.


6-3 Polynomials

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

Remember!


6-3 Polynomials

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


c. 2c3



6-3 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.


6-3 Polynomials

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


6-3 Polynomials



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



–6: degree 0

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


6-3 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.


6-3 Polynomials

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.


6-3 Polynomials


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.


6-3 Polynomials

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

Remember!

y5 = 1y5


6-3 Polynomials

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.


6-3 Polynomials

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.


6-3 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


6-3 Polynomials

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 – 9

Degree 6 Terms 4

4y6 – 5y3 + 2y – 9 is a

6th-degree polynomial.

C. –2xDegree 1 Terms 1

–2x is a linear monomial.


6-3 Polynomials



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.


6-3 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?

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


6-3 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?

Example 5: Application Continued

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


6-3 Polynomials

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


6-3 Polynomials

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.


6-3 Polynomials

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


6-3 Polynomials

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 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

Holt Algebra 1

6-4 Adding and Subtracting Polynomials6-4 Adding and Subtracting Polynomials

Holt Algebra 1

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz



6-3 Polynomials

Warm UpSimplify each expression by combining like terms.

1. 4x + 2x

2. 3y + 7y

3. 8p – 5p

4. 5n + 6n2

Simplify each expression.

5. 3(x + 4)

6. –2(t + 3)

7. –1(x2 – 4x – 6)

6x

10y

3p

not like terms

3x + 12

–2t – 6

–x2 + 4x + 6

Holt Algebra 1

6-4 Adding and Subtracting Polynomials

Add and subtract polynomials.

Objective

Holt Algebra 1


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

Holt Algebra 1


Add or subtract.

Example 1: Adding and Subtracting Monomials

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.

Holt Algebra 1


Add or subtract.

Example 1: Adding and Subtracting Monomials

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.

Holt Algebra 1


Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7.

Remember!

Holt Algebra 1



a. 2s2 + 3s2 + s

Add or subtract.

2s2 + 3s2 + s

5s2 + s

b. 4z4 – 8 + 16z4 + 2

4z4 – 8 + 16z4 + 2

4z4 + 16z4 – 8 + 2

20z4 – 6


Combine like terms.



Holt Algebra 1



c. 2x8 + 7y8 – x8 – y8

Add or subtract.

2x8 + 7y8 – x8 – y8

2x8 – x8 + 7y8 – y8

x8 + 6y8

d. 9b3c2 + 5b3c2 – 13b3c2

9b3c2 + 5b3c2 – 13b3c2

b3c2


Combine like terms.



Holt Algebra 1


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

= 7x2 + 9x + 3

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

7x2 + 9x + 3

Holt Algebra 1


Add.

Example 2: Adding Polynomials

A. (4m2 + 5) + (m2 – m + 6)

(4m2 + 5) + (m2 – m + 6)

(4m2 + m2) + (–m) +(5 + 6)

5m2 – m + 11


Group like terms together.

Combine like terms.

B. (10xy + x) + (–3xy + y)

(10xy + x) + (–3xy + y)

(10xy – 3xy) + x + y

7xy + x + y



Combine like terms.

Holt Algebra 1


Add.

Example 2C: Adding Polynomials

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)


Combine like terms in the second polynomial.

Combine like terms.

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

(6x2 – 4y) + (–5x2 + y)

(6x2 –5x2) + (–4y + y)

x2 – 3ySimplify.

Holt Algebra 1


Add.

Example 2D: Adding Polynomials



Combine like terms.

Holt Algebra 1



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

(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)

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

12a3 + 15a2 – 16a



Combine like terms.

Holt Algebra 1


To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:

–(2x3 – 3x + 7)= –2x3 + 3x – 7

Holt Algebra 1


Subtract.

Example 3A: Subtracting Polynomials

(x3 + 4y) – (2x3)

(x3 + 4y) + (–2x3)

(x3 + 4y) + (–2x3)

(x3 – 2x3) + 4y

–x3 + 4y

Rewrite subtraction as addition of the opposite.



Combine like terms.

Holt Algebra 1


Subtract.

Example 3B: Subtracting Polynomials

(7m4 – 2m2) – (5m4 – 5m2 + 8)

(7m4 – 2m2) + (–5m4 + 5m2 – 8)

(7m4 – 5m4) + (–2m2 + 5m2) – 8

(7m4 – 2m2) + (–5m4 + 5m2 – 8)

2m4 + 3m2 – 8




Combine like terms.

Holt Algebra 1


Subtract.

Example 3C: Subtracting Polynomials

(–10x2 – 3x + 7) – (x2 – 9)

(–10x2 – 3x + 7) + (–x2 + 9)

(–10x2 – 3x + 7) + (–x2 + 9)

–10x2 – 3x + 7–x2 + 0x + 9

–11x2 – 3x + 16



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

Holt Algebra 1


Subtract.

Example 3D: Subtracting Polynomials

(9q2 – 3q) – (q2 – 5)

(9q2 – 3q) + (–q2 + 5)

(9q2 – 3q) + (–q2 + 5)

9q2 – 3q + 0+ − q2 – 0q + 5

8q2 – 3q + 5


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

placeholders.

Combine like terms.

Holt Algebra 1



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



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

Combine like terms.

Holt Algebra 1


A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of plot A can be represented by 3x2 + 7x – 5 and the area of plot B can be represented by 5x2 – 4x + 11. Write a polynomial that represents the total area of both plots of land.

Example 4: Application

(3x2 + 7x – 5)(5x2 – 4x + 11)

8x2 + 3x + 6

Plot A.Plot B.

Combine like terms.

+

Holt Algebra 1



The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant.

Use the information above to write a polynomial that represents the total profits from both plants.

–0.03x2 + 25x – 1500 Eastern plant profit.

–0.02x2 + 21x – 1700 Southern plant profit.Combine like terms.

+–0.05x2 + 46x – 3200

Holt Algebra 1


Lesson Quiz: Part I

Add or subtract.

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

20d2 – 2d – 7

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

Holt Algebra 1


Lesson Quiz: Part II

6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x2 + 12x + 9, and the area of the second wall is modeled by

36x2 – 12x + 1. Write a polynomial that represents the total area of the two walls.

40x2 + 10

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Holt McDougal Algebra 1 6-3 Polynomials 6-3 Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz