How do you find the zeros of a polynomial function?

Definition

The zeros of a polynomial function are the values of $x$ for which the function equals zero. In other words, for a polynomial $f(x)$, the zeros are the solutions to the equation:

$$ f(x) = 0 $$

Finding the zeros involves solving this equation, which can be done by factoring, using the quadratic formula, or other algebraic methods depending on the degree of the polynomial.

Worked Example

Find the zeros of the polynomial:
$$ f(x) = x^2 - 5x + 6 $$

Step 1: Set the function equal to zero
$$ x^2 - 5x + 6 = 0 $$

Step 2: Factor the quadratic
$$ x^2 - 5x + 6 = (x - 2)(x - 3) $$

Step 3: Set each factor equal to zero and solve

• $x - 2 = 0 implies x = 2$
• $x - 3 = 0 implies x = 3$

Conclusion:
The zeros of $f(x)$ are $x = 2$ and $x = 3$.

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Takeaways

• Zeros of a polynomial are the $x$-values where $f(x) = 0$.
• Factoring is a common method for finding zeros, especially for quadratics.
• For higher-degree polynomials, use factoring, synthetic division, or formulas as appropriate.