A polynomial function of degree n can be described by an equation of the form Polynomial P(x) = + + + + x + , … Function where the coefficients , , , represent real numbers, is not zero, and n represents a …, nonnegative integer. Example 1: What are the degree and leading coefficient of 3 2 7 + ? – – Rewrite the expression so the powers of are in decreasing x + + 3 7 –2 – This is a polynomial in one variable. The degree is 4, and the leading coefficient is –2. Example 2: Find if = + 2 10x + 20. f(–5) f(x) – = + 2 10x + 20 f(x) – Original function = + 10(–5) + f(–5) – x with –5. = + 50 + 50 + 20 –125 Evaluate. = –5 Simplify. Example 3: Find 1) if = + 3x 4. g( g(x) – – = + 3x 4 g(x) – Original function 1) = + 3( 1) 4 g( – – – Replace x with 1. – = 2 + 1 + 3 3 4 – – – Evaluate. = + 6 – Simplify. Exercises State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 3 + 6 + 12 2. 100 5 + 10 3. 4 + 6 + 8 10 + 20 – – – 4. 4 3xy + 16 5. 8 9 + 4 36 6. + – – – – – Find and for each function. f(2) f(–5) 7. = 9 8. = 4 3 + 2x 1 9. = 9 4 + 5x + 7 f(x) – f(x) – – f(x) – 5-3 Guide (continued) Polynomial Functions Graphs of Polynomial Functions If the degree is even and the leading coefficient is positive, then f(x) +∞ as x f(x) +∞ as x +∞ → → –∞ → → End Behavior of Polynomial If the degree is even and the leading coefficient is negative, then Functions f(x) as x f(x) as x +∞ → –∞ → –∞ → –∞ → If the degree is odd and the leading coefficient is positive, then
Polynomial Function A polynomial function of degree n can be described by an equation of the form P(x) = + + … + + x + , where the coefficients , , , …, represent real numbers, is not zero, and n represents a nonnegative integer. Example 1: What are the degree and leading coefficient of 3 – 2 – 7 + ? Rewrite the expression so the powers of x are in decreasing –2 + + 3 – 7 This is a polynomial in one variable. The degree is 4, and the leading coefficient is –2. Example 2: Find f(–5) if f(x) = + 2 – 10x + 20. f(x) = + 2 – 10x + 20 Original function f(–5) = + – 10(–5) + x with –5. = –125 + 50 + 50 + 20 Evaluate. = –5 Simplify. Example 3: Find g( – 1) if g(x) = + 3x – 4. g(x) = + 3x – 4 Original function g( – 1) = + 3( – 1) – 4 Replace x with – 1. = – 2 + 1 + 3 – 3 – 4 Evaluate. = + – 6 Simplify. Exercises State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 3 + 6 – + 12 2. 100 – 5 + 10 3. 4 + 6 + 8 – 10 + 20 4. 4 – 3xy + 16 5. 8 – 9 + 4 – 36 6. – + – Find f(2) and f(–5) for each function. 7. f(x) = – 9 8. f(x) = 4 – 3 + 2x – 1 9. f(x) = 9 – 4 + 5x + 7 5-3 Guide (continued) Polynomial Functions Graphs of Polynomial Functions End Behavior of Polynomial Functions If the degree is even and the leading coefficient is positive, then f(x) → +∞ as x → –∞ f(x) → +∞ as x → +∞ If the degree is even and the leading coefficient is negative, then f(x) → –∞ as x → –∞ f(x) → –∞ as x → +∞ If the degree is odd and the leading coefficient is positive, then
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