Polynomial Functions Math Example 4

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Example 4

hard
Sketch the end behavior and find all real zeros of f(x)=−2x3+6x2+8xf(x) = -2x^3 + 6x^2 + 8x. State the multiplicity of each zero.

Solution

  1. 1
    Factor out −2x-2x: f(x)=−2x(x2−3x−4)=−2x(x−4)(x+1)f(x) = -2x(x^2 - 3x - 4) = -2x(x - 4)(x + 1).
  2. 2
    Zeros: x=0x = 0, x=4x = 4, x=−1x = -1, each with multiplicity 1 (all linear factors).
  3. 3
    End behavior: Leading term is −2x3-2x^3. Odd degree, negative leading coefficient → as x→−∞x \to -\infty, f(x)→+∞f(x) \to +\infty; as x→+∞x \to +\infty, f(x)→−∞f(x) \to -\infty.

Answer

Zeros: x=−1,0,4 (each multiplicity 1). Rises left, falls right.\text{Zeros: } x = -1, 0, 4 \text{ (each multiplicity 1). Rises left, falls right.}
For polynomials, factor completely to find zeros and their multiplicities. The leading term alone determines end behavior: the sign of the leading coefficient and whether the degree is odd or even.

About Polynomial Functions

A polynomial function is formed by adding terms of the form axnax^n where nn is a non-negative integer. The highest power determines the degree, which controls the graph's end behavior, maximum turning points, and number of possible real zeros.

Learn more about Polynomial Functions →

More Polynomial Functions Examples

Example 1 easy

Find the degree and leading coefficient of [formula].

Example 2 medium

Find all zeros of [formula].

Example 3 medium

Perform polynomial long division: [formula].

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