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8.07 Polynomial Functions

Zeros of a Polynomial Function

·  A ______of a polynomial function is a value for x that makes f(x) = 0.

·  Factor the polynomial.

·  Set factors equal to zero due to the ______.

·  Solve for each factor.

·  These are the ______.

·  For a factored polynomial, f(x) = (x – a)(x – b)(x – c), the zeros are x = a, x = b, and x = c. The x-intercepts are (a, 0), (b, 0), and (c, 0)

Even versus Odd Degree Functions

·  If a polynomial function has an ______degree, then the ends of the graph will continue in the same direction.

·  If a polynomial function has an ______numbered degree, then the ends of the graph will continue in opposite directions.

End Behavior and Leading Coefficient

End Behavior of a Graph of a Polynomial Function
Odd Degree / Even Degree
The ends will travel in opposite directions. / The ends will travel in the same direction.
Positive Leading Coefficent / Negative Leading Coefficient / Positive Leading Coefficient / Negative Leading Coefficient
Left side continues down, right side continues up.
/ Left side continues up, right side continues down.
/ Both sides continue up.
/ Both sides continue down.

Axis of Symmetry for Parabola

·  The axis of symmetry is a line that divides a parabola into two equal parts. With quadratic functions (polynomial function with degree of 2), the axis of symmetry is a vertical line found by the formula below.

x = -b/2a

Vertex of a Parabola

·  The vertex is the highest point on a parabola that opens downward, and the lowest point on a parabola that opens upward. It also lies on the axis of symmetry.

·  To find the vertex, substitute the value of the axis of symmetry into the function for x.

Example 1: 3x2 = 12x

Steps / Work
·  Step 1 – Set equation equal to zero
·  Step 2 - Factor
·  Step 3 – Set factors equal to zero

Example 2: x2 – 7x + 12 = 0

Steps / Work
·  Step 1 – Set equation equal to zero
·  Step 2 - Factor
·  Step 3 – Set factors equal to zero

Example 3: 6x2 – 9x = 60

Steps / Work
·  Step 1 – Set equation equal to zero
·  Step 2 - Factor
·  Step 3 – Set factors equal to zero