How to Form a Quadratic Polynomial from Its Zeroes

What is the formula?

A monic quadratic polynomial with zeroes α and β is x² − (α + β)x + αβ. Equivalently, expand (x − α)(x − β). Any nonzero constant multiple has the same zeroes, so a question asking for a quadratic polynomial usually has more than one valid scaled answer.

Example: zeroes 3 and −2

Their sum is 1 and their product is −6. The monic polynomial is x² − x − 6. Check by factorising: (x − 3)(x + 2). Substituting x = 3 gives 9 − 3 − 6 = 0; substituting x = −2 gives 4 + 2 − 6 = 0.

The expression 2x² − 2x − 12 also works because it is twice the same polynomial. However, multiplying by zero would produce the zero polynomial, not a quadratic. A specified leading coefficient removes this scaling freedom.

Example: only the sum and product are given

If the sum is −4 and product is 3, substitute directly: x² − (−4)x + 3 = x² + 4x + 3. Factorisation gives (x + 1)(x + 3), whose zeroes −1 and −3 have the required sum and product. You do not have to find the individual zeroes before forming the polynomial.

Fractional zeroes

For zeroes 1/2 and 3, the monic form is x² − (7/2)x + 3/2. Multiplying all terms by 2 gives the convenient integer-coefficient form 2x² − 7x + 3. Multiplying only one term would change the polynomial's roots.

Practice

1. Zeroes 4 and 5: x² − 9x + 20.
2. Sum 2 and product −8: x² − 2x − 8; its zeroes are 4 and −2.
3. Zeroes −2 and −2: x² + 4x + 4. A repeated zero is allowed.

Keep the minus sign before the sum. The product itself retains its own sign.

Need help with your own attempt? Ask a study question. AI answers require an account and paid credits. Check important steps with your textbook.