4.3 S KM & PP 1

x3 5x

472 xx

3223 xxx

POLYNOMIALS

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The word “Polynomial” means “many names” or “many terms”.

A “term” is a “monomial” and has the following form:

a is a Real Number (constant) n is a non-negative integer.

nax

What is a Polynomial?

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A Polynomial is one or more monomials

(terms) combined by addition or subtraction.

Here’s an example:

A Polynomial is...

25x x3 1037x

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Standard Form for a Polynomial is...

Usually, polynomials are written in

STANDARD FORMwhere the terms are

listed so that the powers of the variable are

decending (largest to smallest.) 25xx3 10 37x

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Polynomial Classification by the Number of Terms.

A MONOMIAL is a single term.

A POLYNOMIAL is composed of the sum or difference of TERMS.

A BINOMIAL has two terms.

A TRINOMIAL has three terms.

A POLYNOMIAL with more than three terms is just called a

POLYNOMIAL.

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Polynomial Classification by the

Degree of the Leading Term

The exponent of the variable in the leading term is the DEGREE

of the POLYNOMIAL.

Once a POLYNOMIAL is arranged in STANDARD FORM, the term

with the largest exponent on the variable is called the LEADING

TERM.

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Polynomial Classification:Degree Names

Degree 0 is called CONSTANT.

Degree 1 is called LINEAR.

Degree 2 is called QUADRATIC.

Degree 3 is called CUBIC.

Degree 4 is called 4th degree, and so on for the higher powers.

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72x

Monomial or Term

Examples of a Monomial:

x42x

5

nax

a=2 and n = 7

a=-4 and n = 1

a=1 and n = 2

a=5 and n = 0

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72x

Monomial or Term

In words:

x42x

0

5

nax

The coefficient is 2 and the degree is 7.

The coefficient is -4 and the degree is 1.

The coefficient is 1 and the degree is 2.

The coefficient is 0 and the degree is 0.

The coefficient is 5 and the degree is 0.

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Polynomial Example:-3x+10

x3 10

-3x+10 is a binomialbecause it has two terms.

-3x+10 is the same as -3x1+10x0

x3

10

is the “linear” term.The coefficient is -3. The degree is 1.

is the “constant” termThe degree is 0.The coefficient is 10.

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25x

Polynomial Example:5x2 – 3x + 10

x3 10

5x2 – 3x + 10 is a trinomial because it has three terms.

x3

10

is the “linear” term.The degree is 1.The coefficient is -3.

is the “constant” term.The degree is 0.The coefficient is 10.

25xis the “quadratic” term.The degree is 2.The coefficient is 5.

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25x

Polynomial Example:7x3 + 5x2 – 3x + 10

x3 107x3 + 5x2 – 3x + 10 is a

polynomial.

x3

10

is the “linear” term.The degree is 1.The coefficient is -3.

is the “constant” term.The degree is 0.The coefficient is 10.

25xis the “quadratic” term.The degree is 2.The coefficient is 5.

37x37x

is the “cubic” term.The degree is 3.The coefficient is 7.

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Let’s Arrange and Analyze this Polynomial

xx 357 2 25x x37

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Simplifying a Polynomial

7391158 22 xxxx22 98 xx

432 xx

2x

xx 35 x2

711 4

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Simplify Another Polynomial

653254 22 xxxx22 34 xx

802 xx

2xxx 55 x0

62 8

82 x

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How to EVALUATE aPOLYNOMIAL

To EVALUATE a polynomial we substitute for the variable with the

given number or expression.

(Replace the variable with parentheses and

substitute. 135 2 xx

135 2

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EVALUATE -5x2 + 3x – 1 for x = 2

135 2 xx

135 2

12325 2

12345 1620

15

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EVALUATE -5x2 + 3x – 1 for x = 0

135 2 xx

135 2

10305 2

10305 100

1

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EVALUATE -5x2 + 3x – 1 for x = -

2

135 2 xx

135 2

12325 2

12345 1620

27

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One for Calculus?

23 x

23 x

23 )xx(

233 xx

xEvaluate 3x -2 for x = x+Δx

23

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That’s All for Now!