polynomials and polynomial functions section 5.3
TRANSCRIPT
Polynomials and Polynomial Functions
Section 5.3
Overview
• 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 (example: )
4or
7 32
c
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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
• Number (example: 1, 0, -2, 12, ⅓)
• Variable (example: x, y, z)
• Product of numbers and/or variables (example: 3a2b4, 2y, 5x2)
• Quotient of numbers and/or variables (example: )
• Monomial or Not?
• Yes
• Yes
• Yes
• No
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7 32
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ba
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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 expressionP(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
PolynomialsRead Section 5.4