Polynomial remainder and factor theorems

Dean54321

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Apr 6, 2021
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When P(x) is divided by x-1, the remainder is 1, and P(x) is divided by x+2, the remainder is -8. Find the remainder, if P(x) is divided by (x-1)(x+2).

Is there a rule for this? because we don't know the polynomial P(x).
 

lex

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Mar 3, 2021
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(If P(x) is linear, then you can't divide by \(\displaystyle (x-1)(x+2)\) and get a remainder, so we'll assume they meant to say that P(x) is at least quadratic).

\(\displaystyle P(x)=(x-1)(x+2)Q(x)+R(x) \quad\) (1)
where R(x) is \(\displaystyle ax+b\)
(Note you are dividing by a quadratic, so the remainder may be linear).

Now substitute \(\displaystyle x=1\) and \(\displaystyle x=-2\) into equation (1) and use the facts that you know: \(\displaystyle P(1)=1\) and \(\displaystyle P(-2)=-8\), to find \(\displaystyle a\) and \(\displaystyle b\).
 
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