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

2. Warm Up Evaluate each expression for the given value of x.
1. 2x + 3; x = x2 + 4; x = –3
3. –4x – 2; x = –1 4. 7x2 + 2x; x = 3
Identify the coefficient in each term.
5. 4x y3
7. 2n –s4 7 13 2 69 4 1 2 –5

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

4. Vocabulary monomial degree of a monomial polynomial
degree of a polynomial
standard form of a
leading coefficient
quadratic
cubic
binomial
trinomial

5. 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. 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: = 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.

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

8. Check It Out! Example 1
Find the degree of each monomial. a.
1.5k2m
The degree is 3.
Add the exponents of the variables: = 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.

9. 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.

10. Example 2: Finding the Degree of a Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
11x7: degree 7 3x3: degree 3
Find the degree of each term.
The degree of the polynomial is the greatest degree, 7. B.
:degree 3
:degree 4
–5: degree 0
Find the degree of each term.
The degree of the polynomial is the greatest degree, 4.

11. Check It Out! Example 2
Find the degree of each polynomial.
a. 5x – 6
5x: degree 1
–6: degree 0
Find the degree of each term.
The degree of the polynomial is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
Find the degree of each term.
x3y2: degree 5
x2y3: degree 5
–x4: degree 4
2: degree 0
The degree of the polynomial is the greatest degree, 5.

12. The terms of a polynomial may be written in any order
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.

13. Example 3A: Writing Polynomials in Standard Form
Write the polynomial in standard form. Then give the leading coefficient.
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
–7x5 + 4x2 + 6x + 9.
The standard form is
The leading
coefficient is –7.

14. Example 3B: Writing Polynomials in Standard Form
Write the polynomial in standard form. Then give the leading coefficient.
y2 + y6 – 3y
Find the degree of each term. Then arrange them in descending order:
y2 + y6 – 3y
y6 + y2 – 3y
Degree 2 6 1
The standard form is
The leading
coefficient is 1.
y6 + y2 – 3y.

15. A variable written without a coefficient has a coefficient of 1.
Remember!
y5 = 1y5

16. Check It Out! Example 3a
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 2 5 3
The standard form is
The leading
coefficient is 1.
x5 + 9x3 – 4x

17. Check It Out! Example 3b
Write the polynomial in standard form. Then give the leading coefficient.
18y5 – 3y8 + 14y
Find the degree of each term. Then arrange them in descending order:
18y5 – 3y8 + 14y
–3y8 + 18y5 + 14y
Degree 5 8 1
The standard form is
The leading
coefficient is –3.
–3y8 + 18y5 + 14y.

18. Some polynomials have special names based on their degree and the number of terms they have.
Monomial
Binomial
Trinomial
Polynomial
4 or more 1 2 3 1 2
Constant
Linear
Quadratic 3 4 5
6 or more
6th,7th,degree and so on
Cubic
Quartic
Quintic

19. Example 4: Classifying Polynomials
Classify each polynomial according to its degree and number of terms.
A. 5n3 + 4n
5n3 + 4n is a cubic binomial.
Degree 3 Terms 2
B. 4y6 – 5y3 + 2y – 9
4y6 – 5y3 + 2y – 9 is a
6th-degree polynomial.
Degree 6 Terms 4
C. –2x
–2x is a linear monomial.
Degree 1 Terms 1

20. Check It Out! Example 4
Classify each polynomial according to its degree and number of terms.
a. x3 + x2 – x + 2
x3 + x2 – x + 2 is a cubic polynomial.
Degree 3 Terms 4
b. 6
6 is a constant monomial.
Degree 0 Terms 1
–3y8 + 18y5 + 14y is an 8th-degree trinomial.
c. –3y8 + 18y5 + 14y
Degree 8 Terms 3

21. Example 5: Application
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 –16t , where t is time in seconds. How far above the water will the lip balm be after 3 seconds?
Substitute the time for t to find the lip balm’s height.
–16t
–16(3)
The time is 3 seconds.
–16(9) + 220
Evaluate the polynomial by using the order of operations. – 76

22. Example 5: Application Continued
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 –16t , where t is time in seconds. How far above the water will the lip balm be after 3 seconds?
After 3 seconds the lip balm will be 76 feet from the water.

23. 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.
–16t t + 6
–16(5) (5) + 6
The time is 5 seconds.
–16(25) + 400(5) + 6
Evaluate the polynomial by using the order of operations. – –
1606

24. 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.

25. Lesson 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. 24g g5 – g2
4. 14 – x4 + 3x2 5 4
7g5 + 24g3 – g2 + 10; 7
–x4 + 3x2 + 14; –1

26. Lesson 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 v2 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? ft

27. Adding and Subtracting Polynomials 6-4
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

28. Warm Up Simplify 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

29. Objective
Add and subtract polynomials.

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

31. Example 1: Adding and Subtracting Monomials
Add or subtract.
A. 12p3 + 11p2 + 8p3
12p3 + 11p2 + 8p3
Identify like terms.
Rearrange terms so that like terms are together.
12p3 + 8p3 + 11p2
20p3 + 11p2
Combine like terms.
B. 5x2 – 6 – 3x + 8
Identify like terms.
5x2 – 6 – 3x + 8
Rearrange terms so that like terms are together.
5x2 – 3x + 8 – 6
5x2 – 3x + 2
Combine like terms.

32. Example 1: Adding and Subtracting Monomials
Add or subtract.
C. t2 + 2s2 – 4t2 – s2
t2 + 2s2 – 4t2 – s2
Identify like terms.
Rearrange terms so that like terms are together.
t2 – 4t2 + 2s2 – s2
–3t2 + s2
Combine like terms.
D. 10m2n + 4m2n – 8m2n
10m2n + 4m2n – 8m2n
Identify like terms.
6m2n
Combine like terms.

33. 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!

34. Check It Out! Example 1
Add or subtract.
a. 2s2 + 3s2 + s
2s2 + 3s2 + s
Identify like terms.
5s2 + s
Combine like terms.
b. 4z4 – z4 + 2
4z4 – z4 + 2
Identify like terms.
Rearrange terms so that like terms are together.
4z4 + 16z4 – 8 + 2
20z4 – 6
Combine like terms.

35. Check It Out! Example 1
Add or subtract.
c. 2x8 + 7y8 – x8 – y8
Identify like terms.
2x8 + 7y8 – x8 – y8
Rearrange terms so that like terms are together.
2x8 – x8 + 7y8 – y8
x8 + 6y8
Combine like terms.
d. 9b3c2 + 5b3c2 – 13b3c2
9b3c2 + 5b3c2 – 13b3c2
Identify like terms.
b3c2
Combine like terms.

36. 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
7x2 + 9x + 3
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3

37. Example 2: Adding Polynomials
A. (4m2 + 5) + (m2 – m + 6)
(4m2 + 5) + (m2 – m + 6)
Identify like terms.
Group like terms together.
(4m2 + m2) + (–m) +(5 + 6)
5m2 – m + 11
Combine like terms.
B. (10xy + x) + (–3xy + y)
(10xy + x) + (–3xy + y)
Identify like terms.
Group like terms together.
(10xy – 3xy) + x + y
7xy + x + y
Combine like terms.

38. Example 2C: Adding Polynomials
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
Identify like terms.
Combine like terms in the second polynomial.
(6x2 – 4y) + (–5x2 + y)
(6x2 –5x2) + (–4y + y)
Combine like terms.
x2 – 3y
Simplify.

39. Example 2D: Adding Polynomials
Identify like terms.
Group like terms together.
Combine like terms.

40. Check It Out! Example 2
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)
Identify like terms.
(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a)
Group like terms together.
12a3 + 15a2 – 16a
Combine like terms.

41. 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

42. Example 3A: Subtracting Polynomials
(x3 + 4y) – (2x3)
Rewrite subtraction as addition of the opposite.
(x3 + 4y) + (–2x3)
(x3 + 4y) + (–2x3)
Identify like terms.
(x3 – 2x3) + 4y
Group like terms together.
–x3 + 4y
Combine like terms.

43. Example 3B: Subtracting Polynomials
(7m4 – 2m2) – (5m4 – 5m2 + 8)
(7m4 – 2m2) + (–5m4 + 5m2 – 8)
Rewrite subtraction as addition of the opposite.
(7m4 – 2m2) + (–5m4 + 5m2 – 8)
Identify like terms.
Group like terms together.
(7m4 – 5m4) + (–2m2 + 5m2) – 8
2m4 + 3m2 – 8
Combine like terms.

44. Example 3C: Subtracting Polynomials
(–10x2 – 3x + 7) – (x2 – 9)
(–10x2 – 3x + 7) + (–x2 + 9)
Rewrite subtraction as addition of the opposite.
(–10x2 – 3x + 7) + (–x2 + 9)
Identify like terms.
–10x2 – 3x + 7
–x2 + 0x + 9
Use the vertical method.
Write 0x as a placeholder.
–11x2 – 3x + 16
Combine like terms.

45. Example 3D: Subtracting Polynomials
(9q2 – 3q) – (q2 – 5)
Rewrite subtraction as addition of the opposite.
(9q2 – 3q) + (–q2 + 5)
(9q2 – 3q) + (–q2 + 5)
Identify like terms.
Use the vertical method.
9q2 – 3q + 0
+ − q2 – 0q + 5
Write 0 and 0q as placeholders.
8q2 – 3q + 5
Combine like terms.

46. Check It Out! Example 3
Subtract.
(2x2 – 3x2 + 1) – (x2 + x + 1)
Rewrite subtraction as addition of the opposite.
(2x2 – 3x2 + 1) + (–x2 – x – 1)
(2x2 – 3x2 + 1) + (–x2 – x – 1)
Identify like terms.
Use the vertical method.
–x2 + 0x + 1
+ –x2 – x – 1
Write 0x as a placeholder.
–2x2 – x
Combine like terms.

47. Example 4: Application
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 Write a polynomial that represents the total area of both plots of land.
(3x2 + 7x – 5)
Plot A. +
(5x2 – 4x + 11)
Plot B.
8x2 + 3x + 6
Combine like terms.

48. Check It Out! Example 4
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.
–0.05x2 + 46x – 3200
Combine like terms.

49. 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)
11m2 + 3m
(–4r2 – 3s2)
18pq – 2p
20d2 – 2d – 7
5. (2.5ab + 14b) – (–1.5ab + 4b)
4ab + 10b

50. 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