Usually to find a remainder of a smaller degree polynomial I just divide the two. But, how should I go about finding remainder of division of two larger degree polynomials. In this specific case: $P(x)=x^{100}+2x^{99}-3x^{2}+2x+5$, divided by $Q(x)=x^2+x-2$.
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You should know that P(x)=Q(x)⋅K(x)+R(x) where K(x) is some polynomial and R(x) is the remainder and is of degree strictly less than Q's degree. Suppose that q1 is a root of Q. What is Q(q1) then? What is P(q1)? R(q1)? What about the other root of Q? Is that enough information to finish?
$x^2+x-2 = (x+2)(x-1)$
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$P(1)=1^{100}+2\cdot 1^{99}-3\cdot 1^2 + 2\cdot 1 + 5 = 1+2-3+2+5 = 7$ as well as $P(1)=Q(1)\cdot K(1)+R(1)=0 + R(1)$ so we know that $R(1)=7$
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$P(-2) = (-2)^{100}+2\cdot (-2)^{99}-3\cdot (-2)^2+2\cdot (-2)+5 = -12-4+5=-11 = R(-2)$
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We can then combine those final two pieces of information to get our final answer:
$R(x) = 6x+1$
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1$\begingroup$ Note that this happened to work for this particular problem because of the convenience of how the arithmetic worked out and how low of a degree the polynomial $Q$ was. For most questions asked in academic settings like this, it will usually work out nicely and the question is asking you to spot some nice convenient colorful pattern like this, here noting how the roots of $Q$ will behave. The general problem can be much more difficult. $\endgroup$ Jan 11 at 18:50