2007 iTest Problems/Problem 18
Problem
Suppose that is a cubic with a double root at and another root at b, where and are real numbers. If and , what is ?
Solutions
Solution 1
By Vieta's Formulas, From the first equation, . Substitute into the second equation to get Thus, or . If , then , so by Vieta's Formulas, . If , then , so by Vieta's Formulas, . The answer is .
Solution 2
The equation of the polynomial with double root and single root is Expanding the polynomial results in Since and , we can write a system of equations. Solving for results in or . If , then , so . If , then , so . Thus, the answer is .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 17 |
Followed by: Problem 19 | |
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