Section 3.4: Zeros of Polynomials

This section focuses on finding the zeros of polynomial functions using factoring, synthetic division, and the Rational Root Theorem.

Example 1: Factoring to Find Zeros

Find all zeros of \( f(x) = x^3 - 4x^2 + x + 6 \).

Step 1: Factor by grouping: \( x^2(x-4) + 1(x-4) = (x^2 + 1)(x-4) \)

Step 2: Set each factor = 0: x - 4 = 0 → x = 4; x^2 + 1 = 0 → x = i, x = -i

Example 2: Using Synthetic Division

Find zeros of \( g(x) = 2x^3 - 3x^2 - 8x + 12 \) given that x = 2 is a zero.

Step 1: Perform synthetic division with 2 → quotient: 2x^2 + x - 6

Step 2: Factor quadratic: 2x^2 + x - 6 = (2x-3)(x+2)

Step 3: Zeros: x = 2, x = 3/2, x = -2

Example 3: Rational Root Theorem

Find possible rational zeros of \( h(x) = x^3 - 3x^2 - 4x + 12 \).

Step 1: Factors of constant 12: ±1,2,3,4,6,12

Step 2: Factors of leading coefficient 1 → same list

Step 3: Possible rational zeros: ±1,2,3,4,6,12

Practice Problems

  1. Find all zeros of f(x) = x^3 + 2x^2 - x - 2
  2. Given x = -1 is a zero, find the remaining zeros of g(x) = x^3 + x^2 -4x -4
  3. List possible rational zeros for h(x) = 2x^3 - 5x^2 + x + 2
  4. Use synthetic division to find zeros of k(x) = x^3 - 6x^2 + 11x - 6, given x = 1 is a zero
  5. Find all zeros of p(x) = x^3 - x^2 - x + 1
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