4.3 S KM & PP 1
x3 5x
472 xx
3223 xxx
POLYNOMIALS
4.3 S KM & PP 2
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?
4.3 S KM & PP 3
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
4.3 S KM & PP 5
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.
4.3 S KM & PP 6
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.
4.3 S KM & PP 7
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.
4.3 S KM & PP 8
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
4.3 S KM & PP 9
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.
4.3 S KM & PP 10
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.
4.3 S KM & PP 11
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.
4.3 S KM & PP 12
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
4.3 S KM & PP 18
EVALUATE -5x2 + 3x – 1 for x = 0
135 2 xx
135 2
10305 2
10305 100
1
4.3 S KM & PP 19
EVALUATE -5x2 + 3x – 1 for x = -
2
135 2 xx
135 2
12325 2
12345 1620
27
4.3 S KM & PP 20
One for Calculus?
23 x
23 x
23 )xx(
233 xx
xEvaluate 3x -2 for x = x+Δx
23
4.3 S KM & PP 21
That’s All for Now!