5.3 Polynomial Functions

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

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

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)

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)

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)

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)

End Behavior: Even Positive Negative Odd Degree Leading Coefficient Left-hand Behavior Right-hand Behavior Example Picture Even Positive Negative Odd

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

Real Zeros The x-intercepts of the function.

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 ________.

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.

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.