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This text provides examples and guided practice exercises for evaluating polynomial expressions using direct and synthetic substitution, as well as identifying polynomial functions based on their standard form, degree, type, and leading coefficient. It also covers the end behavior of even and odd functions.
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Warm Up #8 Evaluate the expression when x=–4 2.–3x3 – 2x2 + 10 1.x2 + 5x (-4)2 + 5(-4) –3(-4)3 – 2(-4)2 + 10 16 – 20 192– 32 + 10 -4 170
1 1. h (x) = x4 – x2 + 3 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. Ask yourself – are the exponents all whole numbers??? Are the coefficients real. Yes, in standard form, deg 4, type quartic, LC 1 4 3. f (x) = 5x2 + 3x –1– x No
2. g (x) = 7x – 3 + πx2 for Examples 1 GUIDED PRACTICE Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. 4. k (x) = x + 2x– 0.6x5 ANSWER ANSWER Yes, No Deg: 2, type: quadratic, LC: π
EXAMPLE 2 Evaluate by direct substitution Use direct substitution to evaluate5. f (x) = 2x4 – 5x3 –4x + 8whenx = 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 – 12 + 8 Evaluate powers and multiply. = 23 Simplify
for Example 2 GUIDED PRACTICE Use direct substitution to evaluate the polynomial function for the given value of x. 6. f (x) = x4 + 2x3 + 3x2 – 7; x = –2 f (x) = (-2)4 + 2(-2)3 + 3(-2)2 – 7 f (x) = 16 – 16 + 12 – 7 f (x) = 5
Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left. STEP 1 EXAMPLE 3 Evaluate by synthetic substitution Use synthetic substitution to evaluate f (x) from Example2 when x = 3. 7. f (x) = 2x4 – 5x3 –4x + 8 The last number is the answer f (3) = 23 9 15 6 3 23 2 1 3 5 STEP 2 Bring down the first coefficient(leading) STEP 3 Multiply by the x-value and add to the next coefficient – continue to the end
) Example 4 (not on paper) Use synthetic substitution to evaluate the polynomial function for the given value of x. 6. f(x) = 5x3 + 3x2–x + 7; x = 2 2 5 3 -1 7 10 26 50 The answer is 57 5 13 25 57
for Examples 3 and 4 GUIDED PRACTICE Use synthetic substitution to evaluate the polynomial function for the given value of x. 8. f (x) = x4+ 2x3+ 3x2 -7; x = -2 -2 1 2 3 0 -7 - 2 0 -6 12 The answer is 5 1 0 3 -6 5
Polynomial End Behavior Even Functions Even Functions Positive Leading Coefficient Right: f(x) +∞ as x +∞ Left: f(x) +∞ as x -∞ Even Functions Negative Leading Coefficient Left: f(x) -∞ as x -∞ Right: f(x) -∞ as x +∞
Polynomial End Behavior Odd Functions Odd Functions Positive Leading Coefficient Left: f(x) -∞ as x -∞ Right: f(x) +∞ as x +∞ Odd Functions Negative Leading Coefficient Left: f(x) +∞ as x -∞ Right: f(x) -∞ as x +∞
ANSWER The correct answer is D. EXAMPLE 4 Standardized Test Practice
10. Describe the degree and leading coefficient and end behavior of the polynomial function whose graph is shown. ANSWER degree: odd, leading coefficient: negative Left: f(x) +∞ as x -∞ Right: f(x) -∞ as x +∞ for Examples 3 and 4 GUIDED PRACTICE
for Examples 5 and 6 GUIDED PRACTICE Graph the polynomial function. 2. f(x) = x4 – x3 – 4x2 + 4 x f(x) -3 76 -2 12 -1 2 0 4 1 0 2 -4 3 22
Classwork Worksheet 5.2 and 5.2.2 finish all in class Textbook Homework Pages 341 – 342 (3- 48) multiples of 3
