Polynomial Functions Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Find all zeros of p(x)=x3−4x2+x+6p(x) = x^3 - 4x^2 + x + 6.

Solution

  1. 1
    Try x=−1x = -1: (−1)3−4(−1)2+(−1)+6=−1−4−1+6=0(-1)^3 - 4(-1)^2 + (-1) + 6 = -1 - 4 - 1 + 6 = 0 ✓
  2. 2
    Divide by (x+1)(x + 1) using synthetic division: x3−4x2+x+6=(x+1)(x2−5x+6)x^3 - 4x^2 + x + 6 = (x + 1)(x^2 - 5x + 6).
  3. 3
    Factor the quadratic: x2−5x+6=(x−2)(x−3)x^2 - 5x + 6 = (x - 2)(x - 3).
  4. 4
    The zeros are x=−1,2,3x = -1, 2, 3.

Answer

x=−1,  x=2,  x=3x = -1, \; x = 2, \; x = 3
Use the Rational Root Theorem to test possible rational zeros, then reduce the degree by synthetic division. A cubic reduces to a quadratic, which you can factor or use the quadratic formula on.

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 3 medium

Perform polynomial long division: [formula].

Example 4 hard

Sketch the end behavior and find all real zeros of [formula]. State the multiplicity of each zero.

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