Forming a quadratic polynomial from its zeros

This lesson shows how to build a quadratic polynomial when its two zeros are known, using the sum and the product of those zeros.

The general form

If $\alpha$ and $\beta$ are the zeros of a quadratic polynomial, then the polynomial can be written as

$x^2 - (\alpha + \beta)x + \alpha\beta.$

The coefficient of $x$ is the sum of the zeros (with a minus sign), and the constant term is the product of the zeros.

Worked example

Suppose the zeros are $\alpha = 2$ and $\beta = 3$. Substitute them into the form above.

Sum of the zeros: $\alpha + \beta = 2 + 3 = 5$.

Product of the zeros: $\alpha\beta = 2 \times 3 = 6$.

So the polynomial is

$x^2 - (2 + 3)x + (2 \times 3) = x^2 - 5x + 6.$

Key takeaways

• A quadratic with zeros $\alpha$ and $\beta$ can be written as $x^2 - (\alpha + \beta)x + \alpha\beta$.
• The middle coefficient comes from the sum of the zeros, and the constant term from their product.
• For zeros $2$ and $3$, this gives $x^2 - 5x + 6$.