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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 x’s 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**

By Degree: Degree Name Example n = 0 constant 3 n = 1 linear 5x + 4 n = 2 quadratic 2x2 + 3x - 2 n = 3 cubic 5x3 + 3x2 – x + 9 n = 4 quartic 3x4 – 2x3 + 8x2 – 6x + 5 n = 5 quintic -2x5 + 3x4 – x3 + 3x2 – 2x + 6 Three Terms: Trinomial 3+ Terms: Polynomial One Term: Monomial Two Terms: Binomial By Number of Terms:

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**Classify each polynomial by degree and by number of terms.**

a) 5x + 2x3 – 2x2 b) x5 – 4x3 – x5 + 3x2 + 4x3 c) x2 + 4 – 8x – 2x3 d) 3x3 + 2x – x3 – 6x5 e) 2x + 5x7 quintic trinomial cubic polynomial Not a polynomial cubic trinomial quadratic monomial 7th degree binomial

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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. a. h (x) = x4 – x2 + 3 1 4 SOLUTION Yes it’s 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 it’s 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) = 5x2 + 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 + 2x – 0.6x5 SOLUTION d. The function is not a polynomial function because the term 2x 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. f (x) = 13 – 2x p (x) = 9x4 – 5x – 2 + 4 not a polynomial function polynomial function; f (x) = –2x + 13; degree 1, type: linear, leading coefficient: –2 h (x) = 6x2 + π – 3x polynomial function; h(x) = 6x2 – 3x + π ; degree 2, type: quadratic, leading coefficient: 6

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**What do we do first to evaluate this?**

EXAMPLE 2 Evaluate by direct substitution Use direct substitution to evaluate f (x) = 2x4 – 5x3 – 4x + 8 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8 Write original function. f (3) = 2(3)4 – 5(3)3 – 4(3) + 8 Substitute 3 for x. = 162 – 135 – What do we do first to evaluate this? = 23 PEMDAS

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**5 9 GUIDED PRACTICE for Examples 1 and 2**

Use direct substitution to evaluate the polynomial function for the given value of x. f (x) = x4 + 2x3 + 3x2 – 7; x = –2 5 g(x) = x3 – 5x2 + 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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**Graph Trend based on Leading Coefficient**

Positive Odd degree—right side up, left side down Even degree—both sides up Negative Odd degree—right side down, left side up Even degree—both sides down

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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 isn’t symmetric around the y axis or origin

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**Tell whether it is even/odd/neither**

f(x)= x2 + 2 f(x)= x2 + 4x f(x)= x3 f(x)= x3 + x f(x)= x3 + 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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An Introduction to Polynomials

An Introduction to Polynomials

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