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5.3

5.3. Polynomial Functions. Polynomial functions-. A polynomial in one variable is a function in the form f(x) = 5x 8 + 3x 7 – 8x 6 + … + 2x 1 + 11 a n is the leading coefficient n is the degree of the polynomial a 0 is the constant term. Type of polynomial. DEGREE 0 1 2

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5.3

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  1. 5.3 Polynomial Functions

  2. Polynomial functions- A polynomial in one variable is a function in the form f(x) = 5x8 + 3x7 – 8x6 + … + 2x1 + 11 • an is the leading coefficient • n is the degree of the polynomial • a0 is the constant term

  3. Type of polynomial DEGREE • 0 • 1 • 2 • 3 • 4 TYPE Constant Linear Quadratic Cubic Quartic

  4. Example 1 : Answer these questions for the following functions. • Is it a polynomial in one variable? Explain. • What is the degree of the polynomial • What is the leading coefficient of the polynomial? • What is the constant (y-intercept) of the polynomial? • What type of polynomial is it? a. f(x) = 3x2 + 3x3 – 7x4 + x – 6 b. f(x, y) = 4x5 – 14xy4 + x – 6 c. f(y) = (5 – 4y)(6 + 2y)

  5. To evaluate a function, you need to plug a value into the function. Example 2: a) f(x) = 3x5 – x4 – 5x + 10 Find f(–2)

  6. To evaluate a function, you need to plug a value into the function. Example 2: b) f(x) = x5 – x4 – 5x + 10 and g(x) = 2x3 – x – 5 Find f(3d)

  7. To evaluate a function, you need to plug a value into the function. Example 2: c) f(x) = x5 – x4 – 5x + 10 and g(x) = 2x3 – x – 5 Find g(p2)

  8. End Behavior:

  9. Example 5: Describe the end behavior for each polynomial. a. f(x) = 3x5 – x4 – 5x + 10 b. f(x) = 3x3 – 4x6 + 2x

  10. Real Zeros • The x-intercepts of the function.

  11. Degree of polynomial • Turning points – the points on a graph where the function changes its vertical direction (up/down). • If the degree of the polynomial is n, there will be at most n – 1 turning points • Ex) If a polynomial has 8 turning points then the degree is ________.

  12. Example 6 Do the following for the graph: a) Describe the end behavior b) Determine if it’s an odd-degree or an even-degree function c) State the number of real zeros. • Answer: • as x → –∞, f(x) → –∞ and as x → +∞, f(x) → –∞ • It is an even-degree polynomial function. • The graph does not intersect the x-axis, so the function has no real zeros.

  13. Example 7 Do the following for the graph: a) Describe the end behavior b) Determine if it’s an odd-degree or an even-degree function c) State the number of real zeros. • Answer: • As x → –∞, f(x) → –∞ and as x → +∞ , f(x) → +∞ • It is an odd-degree polynomial function. • The graph intersects the x-axis at one point, so the function has one real zero.

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