Lesson 44: Zeros of Polynomial Functions
The zeros of a polynomial function are the x-values for which f(x) = 0. Identifying zeros is essential for graphing and solving polynomial equations.
Finding Zeros of Polynomial Functions
Methods to find zeros:
- Factoring: Factor the polynomial and set each factor equal to zero.
- Remainder/Factor Theorem: Test possible rational roots using factors of the constant term and leading coefficient.
- Synthetic Division: Divide by potential zeros to verify.
- Quadratic Formula: For quadratic factors that cannot be easily factored.
Examples
Example 1: f(x) = x³ − 6x² + 11x − 6
- Possible rational zeros: ±1, ±2, ±3, ±6
- Test x = 1 → f(1) = 0 → x − 1 is a factor
- Divide f(x) by x − 1 → quotient: x² − 5x + 6
- Factor x² − 5x + 6 = (x − 2)(x − 3)
- Zeros: x = 1, 2, 3
Example 2: f(x) = 2x³ + 3x² − 2x − 3
- Group terms: (2x³ + 3x²) − (2x + 3) = x²(2x + 3) − 1(2x + 3)
- Factor: (2x + 3)(x² − 1) = (2x + 3)(x − 1)(x + 1)
- Zeros: x = −3/2, 1, −1
Practice Problems
- Find zeros: f(x) = x³ − 4x² − x + 4
- Find zeros: f(x) = x³ + x² − 4x − 4
- Find zeros: f(x) = 2x³ − 3x² − 11x + 6
- Factor completely and find zeros: f(x) = xⴠ− 5x² + 4
- Verify zeros using synthetic division: f(x) = x³ − 7x + 6