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Turn In GHSGT Worksheet!!

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Polynomial Functions 2.1 (M3)

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What is a Polynomial? 1 or more terms Exponents are whole numbers (not a fractional) Coefficients are all real numbers (no imaginary #s) NO xs in the denominator or under the radical It is in standard form when the exponents are written in descending order.

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Classification of a Polynomial DegreeNameExample -2x 5 + 3x 4 – x 3 + 3x 2 – 2x + 6 n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 constant 3 linear 5x + 4 quadratic 2x 2 + 3x - 2 cubic 5x 3 + 3x 2 – x + 9 quartic 3x 4 – 2x 3 + 8x 2 – 6x + 5 quintic One Term: Monomial Two Terms: Binomial Three Terms: Trinomial 3+ Terms: Polynomial

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Classify each polynomial by degree and by number of terms. a) 5x + 2x 3 – 2x 2 cubic trinomial b) x 5 – 4x 3 – x 5 + 3x 2 + 4x 3 quadratic monomial c) x – 8x – 2x 3 d) 3x 3 + 2x – x 3 – 6x 5 cubic polynomial quintic trinomial e) 2x + 5x 7 7 th degree binomial Not a polynomial

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EXAMPLE 1 Identify polynomial functions 4 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h (x) = x 4 – x SOLUTION a.Yes its a Polynomial. It is in standard form. Degree 4 - Quartic Its leading coefficient is 1.

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EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. SOLUTION b. Yes its a Polynomial. Standard form is Degree 2 – Quadratic Leading Coefficient is

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EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. c. f (x) = 5x 2 + 3x –1 – x SOLUTION c. The function is not a polynomial function because the term 3x – 1 has an exponent that is not a whole number.

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EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. d. k (x) = x + 2 x – 0.6x 5 SOLUTION d. The function is not a polynomial function because the term 2 x does not have a variable base and an exponent that is a whole number.

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GUIDED PRACTICE for Examples 1 and 2 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. 1. f (x) = 13 – 2x polynomial function; f (x) = –2x + 13; degree 1, type: linear, leading coefficient: –2 2. p (x) = 9x 4 – 5x – h (x) = 6x 2 + π – 3x polynomial function; h(x) = 6x 2 – 3x + π ; degree 2, type: quadratic, leading coefficient: 6 not a polynomial function

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EXAMPLE 2 Evaluate by direct substitution Use direct substitution to evaluate f (x) = 2x 4 – 5x 3 – 4x + 8 when x = 3. f (x) = 2x 4 – 5x 3 – 4x + 8 f (3) = 2(3) 4 – 5(3) 3 – 4(3) + 8 = 162 – 135 – = 23 Write original function. Substitute 3 for x.

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GUIDED PRACTICE for Examples 1 and 2 Use direct substitution to evaluate the polynomial function for the given value of x. 4. f (x) = x 4 + 2x 3 + 3x 2 – 7; x = – g(x) = x 3 – 5x 2 + 6x + 1; x = 4 9

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Graph Trend based on Degree Even degree - end behavior going the same direction Odd degree – end behavior (tails) going in opposite directions

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Positive –Odd degreeright side up, left side down –Even degreeboth sides up Negative –Odd degreeright side down, left side up –Even degreeboth sides down Graph Trend based on Leading Coefficient

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Symmetry: Even/Odd/Neither First look at degree Even if it is symmetric respect to y-axis –When you substitute -1 in for x, none of the signs change Odd if it is symmetric with respect to the origin –When you substitute -1 in for x, all of the signs change. Neither if it isnt symmetric around the y axis or origin

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Tell whether it is even/odd/neither 1.f(x)= x f(x)= x 2 + 4x 3.f(x)= x 3 4.f(x)= x 3 + x 5.f(x)= x 3 + 5x +1

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Additional Vocabulary to Review Domain: set of all possible x values Range: set of all possible y values Symmetry: even (across y), odd (around origin), or neither Interval of increase (where graph goes up to the right) Interval of decrease (where the graph goes down to the right) End Behavior: f(x) ____ as x + f(x) ______ as x -

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Math 3 Book Page 67 #1 – 4 Page 68 #5 – 7 Page 69 #1 – 5, 8 – 16 a) Classify by degree and # of terms b) Even, Odd or Neither c) End Behavior

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Blue Algebra 2 Books P.429 # 4 –10 –#9 and 10 A) Domain and Range B) Classify by degree and # of terms C) Even, Odd or Neither D) End

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