Sum and Product Roots Lesson 6-5

The Sum and the Product Roots Theorem In a quadratic whose leading coefficient is 1: the sum of the roots is the negative of the coefficient of x; the product of the roots is the constant term.

Sum and Product of Roots If the roots of with are and, then and.

Example 1 Construct the quadratic whose roots are 2 and 3. Solution. The sum of the roots is 5, their product is 6, therefore the quadratic is x² − 5x + 6. The sum of the roots is the negative of the coefficient of x. The product of the roots is the constant term.

Example 2 Construct the quadratic whose roots are 2 +, 2 −. Solution. The sum of the roots is 4. Their product is the Difference of two squares:Difference of two squares 2² − ( )² = 4 − 3 = 1. The quadratic therefore is x² − 4x + 1.

Example 3 Construct the quadratic whose roots are 2 + 3i, 2 − 3i, where i is the complex unit.complex unit The sum of the roots is 4. The product again is the Difference of Two Squares: 4 − 9i² = = 13. The quadratic with those roots is x² − 4x + 13.

Example 4 Construct the quadratic whose roots are −3, 4. The sum of the roots is 1. Their product is −12. Therefore, the quadratic is x² − x − 12.

Example 5 Construct the quadratic whose roots are 3 +, 3 −. The sum of the roots is 6. Their product is 9 − 3 = 6. Therefore, the quadratic is x² − 6x + 6.

Example 6 Construct the quadratic whose roots are 2 + i, 2 − i. The sum of the roots is 4. Their product is 4 − ( i )² = = 9. Therefore, the quadratic is x² − 4x + 9.