Polynomials

The zeroes of a polynomial are where its graph meets the x-axis, and for a quadratic they are tied to the coefficients by two simple rules.

Key ideas

• A zero of p(x) is a number k with p(k) = 0. Graphically, it is where y = p(x) meets the x-axis.
• A polynomial of degree n has at most n zeroes: a linear polynomial one, a quadratic two, a cubic three.
• The graph of a quadratic is a parabola: it opens upwards when a > 0 and downwards when a < 0.
• For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a.
• A quadratic with a given sum and product of zeroes is k[x² − (sum)x + (product)] for any k ≠ 0.
• For a cubic ax³ + bx² + cx + d: the sum of the zeroes is −b/a, the sum of products in pairs is c/a, and the product is −d/a.
• Expressions like α² + β² or 1/α + 1/β can be found from the sum and product without finding α and β.

Remember

Watch out

• Writing the sum of zeroes as b/a instead of −b/a.
• Forgetting to clear fractions, or giving only one of the possible polynomials without the factor k.
• Reading the zeroes off the y-axis instead of the x-axis.

Worked example