Polynomials

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Classify polynomials and write

polynomials in standard form.

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monomial

degree of a monomial

polynomial

degree of a polynomial

standard form of a

polynomial

leading coefficient

Vocabulary

binomial

trinomial

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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

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A

polynomial

is a monomial or a sum or

difference of monomials.

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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

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Student 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.

b. 2c3

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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.

The degree of the polynomial is the greatest degree, 4. :degree 3 :degree 4

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Student Example 2

Find the degree of each polynomial. a. 5x – 6

5x: degree 1

b. x3y2 + x2y3 – x4 + 2

x3y2: degree 5

–6: degree 0

x2y3: degree 5

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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

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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

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Example 3B: Writing Polynomials in Standard Form

y2 + y6 − 3y

y2 + y6 – 3y _y₆₊_y₂_{– 3y}

Degree 2 6 1 6 2 1

The standard form is The leading coefficient is 1. y

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Student Example 3a

16 – 4x2 + x5 + 9x3

16 – 4x2 + x5 + 9x3 _x5 + 9x3 – 4x2 + 16

Degree 0 2 5 3 5 3 2 0

The standard form is The leading

coefficient is 1. x

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Student 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. –3y

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CFU

Find the degree of each polynomial. 1. 25x2 – 3x4

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

2. 14 – x4 + 3x2

4

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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

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

Example 4: Classifying Polynomials

A. 5n3 + 4n

Degree 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. –2x

Degree 1 Terms 1

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Check It Out! Example 4

a. x3 + x2 – x + 2

Degree 3 Terms 4 x

3 + x2 – x + 2 is a

cubic polymial.

b. 6

Degree 0 Terms 1 6 is a constant monomial.

c. –3y8 + 18y5 + 14y Degree 8 Terms 3

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CFU

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Part 1

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Part 2:

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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

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Add.

Example : Adding Polynomials

A. (4m2 + 5) + (m2 – m + 6) (4m2 + 5) + (m2 – m + 6)

(4m2 + m2) + (–m) +(5 + 6) 5m2 – m + 11

Identify like terms. Group like terms

together.

Combine like terms.

B. (10xy + x) + (–3xy + y)

(10xy + x) + (–3xy + y) (10xy – 3xy) + x + y

7xy + x + y

Identify like terms. Group like terms

together.

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Add.

Student Example 1: Adding Polynomials

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

Identify like terms.

Group like terms together within each polynomial.

Combine like terms. (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

(6x2 + 3x2 – 8x2) + (3y – 4y – 2y)

Use the vertical method. 6x2 – 4y

+ –5x2 + y

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Student Example 2: Adding Polynomials

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

(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)

(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a) 12a3 + 15a2 – 16a

Group like terms together.

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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.

Group like terms together.

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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

Rewrite subtraction as

addition of the opposite. Identify like terms.

Group like terms together.

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Subtract.

Student Example 1: 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

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Subtract.

(9q2 – 3q) – (q2 – 5) (9q2 – 3q) + (–q2 + 5)

(9q2 – 3q) + (–q2 + 5)

9q2 – 3q + 0

+ − q2 – 0q + 5

8q2 – 3q + 5

Use the vertical method. Write 0 and 0q as

placeholders.

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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

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Part 2

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Find the degree of each polynomial.

6. 7a3b2 – 2a4 + 4b – 15 Write each polynomial in standard form. Then give the leading coefficient.

5. 24g3 + 10 + 7g5 – g2 Classify each polynomial according to its degree and number of terms.

4. 18x2 – 12x + 5 2. 7m2 + 3m + 4m2

3. (r2 + s2) – (5r2 + 4s2)

1.(14d2 – 8) + (6d2 – 2d +1)

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