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Sum and Product Roots

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.

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Sum and Product Roots

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  1. Sum and Product Roots Lesson 6-5

  2. 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.

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

  4. 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.

  5. 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:  2² − ( )² = 4 − 3 = 1. The quadratic therefore is  x² − 4x + 1.

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

  7. 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.

  8. 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.

  9. 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 = 4 + 5 = 9. Therefore, the quadratic is  x² − 4x + 9.

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