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ax^3 -7x^2 + 7x-2, x^3 -2ax^2 +8x-8. I’m to solve for a, and know that when both equations are divided by x-2 they leave the same remainder. So far I’ve verified through synthetic division that they do indeed leave the same remainder, not helpful, but it’s what I have so far. Some helpful insight would be appreciated.

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LammettHash

Maybe an easier way to do it is to apply the polynomial remainder theorem. It says that dividing a polynomial [tex]p(x)[/tex] by [tex]x-c[/tex] leaves a remainder of [tex]p(c)[/tex].

In this case, [tex]c=2[/tex], and we have

[tex]a(2)^3-7(2)^2+7(2)-2=8a-16[/tex]

[tex](2)^3-2a(2)^2+8(2)-8=16-8a[/tex]

Then

[tex]8a-16=16-8a\implies16a=32\implies a=2[/tex]

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