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

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

Example : 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.

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

Find the degree of each polynomial.

Example : 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

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7-5 Polynomials

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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

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

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

Example : 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.

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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.

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

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

Example: 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.

Holt Algebra 1

7-5 Polynomials


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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

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

7-5 Polynomials

Add or Subtract..

Example: 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

7-5 Polynomials

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

7-5 Polynomials

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a. 2x8 + 7y8 – x8 – y8

Add or subtract.

2x8 + 7y8 – x8 – y8

2x8 – x8 + 7y8 – y8

x8 + 6y8

Identify like terms.Rearrange terms so that

like terms are together.

Combine like terms.

Holt Algebra 1

7-5 Polynomials

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

= 7x2 + 9x + 3

5x2 + 4x + 1

+ 2x2 + 5x + 2

7x2 + 9x + 3

Holt Algebra 1

7-5 Polynomials

Add.

Example: 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

7-5 Polynomials

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

7-5 Polynomials

Subtract.

Example: 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

Rewrite subtraction as addition of the opposite.



Combine like terms.

Holt Algebra 1

7-5 Polynomials

Subtract.

Example: 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

7-5 Polynomials

Check It Out!

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

7-5 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

Holt Algebra 1

7-5 PolynomialsLesson Quiz: Part II


5. 18x2 – 12x + 5 quadratic trinomial

6. 2x4 – 1 quartic binomial

Add or subtract.

7. 7m2 + 3m + 4m2

8. (r2 + s2) – (5r2 + 4s2)

9. (10pq + 3p) + (2pq – 5p + 6pq)

10. (14d2 + 1) + (6d2 – 2d - 8)

11m2 + 3m

(–4r2 – 3s2)

18pq – 2p

20d2 – 2d – 7

Holt Algebra 1

7-5 Polynomials

Lesson Quiz: Part III

Add or subtract.

11. 7m2 + 3m + 4m2

12. (r2 + s2) – (5r2 + 4s2)

13. (10pq + 3p) + (2pq – 5p + 6pq)

14. (14d2 + 1) + (6d2 – 2d - 8)

(–4r2 – 3s2)

11m2 + 3m

18pq – 2p

20d2 – 2d – 7

15. (2ab + 14b) – (–5ab + 4b) 7ab + 10b

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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