Transcript Polynomials and Polynomial Functions

Polynomials and Polynomial
Functions
Section 5.3
Overview
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Terms
Types of Polynomials
Degree and Coefficients
Combining Like Terms
Polynomial Functions
Graphs of Polynomial Functions
Terms
• Number (example: 1, 0, -2, 121)
• Variable (example: x, y, z)
• Product of numbers and/or variables
(example: 3a2b4, 2y, 5x2)
• Quotient of numbers and/or variables
2 3
7
a
b
(example:
)
or
t
4c
Types of Polynomials
• Monomial
– Product of constants or variables
– Variables only raised to whole number
exponents (i.e. 0 or positive integer)
– Example: 7, t, 23x2y, ⅓a5
– Note: Terms like 1/t or x-2 are not monomials
Terms
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• Monomial or Not?
Number (example: 1, 0, -2, 12, ⅓) • Yes
Variable (example: x, y, z)
• Yes
Product of numbers and/or variables • Yes
(example: 3a2b4, 2y, 5x2)
Quotient of numbers
and/or
variables
• No
2 3
(example: 7 or a b )
t
4c
Types of Polynomials
• Polynomial
– A monomial or a sum of monomials
– Example: 4x + 7, ⅓t2, 6a + 7, 6, 0
– When polynomial is sum of monomials, each
monomial is called a term of the polynomial
Types of Polynomials
• ID the terms of the polynomial
3t4 – 5t6 – 4t - 2
Types of Polynomials
• Polynomial with one term
– Monomial
– 4x2
• Two terms
– Binomial
– 2x + 4
• Three terms
– Trinomial
– 3t2 + 4t + 7
• Four terms
– No special name for polynomials with four or more terms
– 4x3 - 5x2 + xy - 8
Degrees and Coefficients
• Degree of a term
– The number of variable factors in that term
– The degree of 7t2 is 2 (t and t)
• Coefficient
– The part of the term that is a constant factor
(i.e. the numeral)
– The coefficient of 3x is 3
Degree and Coefficients
• Leading term – term of highest degree
• Leading coefficient – coefficient of the
leading term
• Degree of the polynomial – degree of the
leading term
• Example: 3x2 – 8x3 + 5x4 + 7x - 6
Combining Like terms
• Like terms (or similar terms)
– Constant terms
– Terms containing the same variable(s) raised to the
same power(s)
• To simplify certain polynomials, you can often
combine, or collect, like terms
– Adding or subtracting like terms
– Write solution in descending order with term of
highest degree first, followed by term of next highest
degree, and so on
Polynomial Functions
• Polynomial function – function involving a
polynomial expression
P(x) = 5x7 + 3x5 – 4x2 -5
• Linear function – degree of polynomial is 1
f(x) = 4x + 5
• Quadratic function – degree is 2
f(x) = 3x2 – 4x + 5
• Cubic function – degree is 3
f(x) = 2x3 + 3x2 – 4x + 5
• Quartic function – degree is 4
f(x) = x4 – 2x3 + 3x2 – 4x + 5
Graphs of Polynomial Functions
• Common characteristics (refer to graphs
on p. 379)
– Smooth line
– Continuous line
– Domain is all real numbers, unless otherwise
specified
• Range
Next up:
Addition And Subtraction of
Polynomials
Read Section 5.4