SOLUTION: What polynomial of lowest degree must be multiplied to 2x^3-5x^2-4x+12 to make it a perfect square?

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Question 702288: What polynomial of lowest degree must be multiplied to 2x^3-5x^2-4x+12 to make it a perfect square?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
It is a factoring and division of polynomials problem
P%28x%29=2x%5E3-5x%5E2-4x%2B12
P%282%29=2%2A2%5E3-5%2A2%5E2-4%2A2%2B12 --> P%282%29=2%2A8-5%2A4-8%2B12 --> P%282%29=16-20-8%2B12 --> P%282%29=0
That means that P%28x%29 is divisible by %28x-2%29
Dividing, we find that P%28x%29=%28x-2%29%282x%5E2-x-6%29

Then, it turns out that 2x%5E2-x-6=%28x-2%29%28x%2B3%29,
so P%28x%29=%28x-2%29%7Bx-2%29%28x%2B3%29,
and %28x%2B3%29%2AP%28x%29=%28x-2%29%28x-2%29%28x%2B3%29%28x%2B3%29 --> highlight%28%28x%2B3%29%29%2AP%28x%29=%28%28x-2%29%28x%2B3%29%29%5E2
The answer is highlight%28x%2B3%29.

How could you factor 2x%5E2-x-6?
If you are good at factoring, you would have no problem.
Otherwise, you could find that the value of 2x%5E2-x-6 for x=2 is 0,
and dividing by %28x-2%29 again, would get the factoring.
Another way to do it, would be solving 2x%5E2-x-6=0 to find that x=2 and x=-3 are the roots of 2x%5E2-x-6.