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Remainder Factor Theorem

You might have heard of remainders in simple division, but what about remainders when dividing polynomials?

Remainder Factor Theorem

         

If \(f(x)=(x-14)(x+1)(x^2+2x+6)\), what is the sum of all real values \(x\) satisfying \(f(x)=0\)?

Consider the polynomial \(f(x)=\frac{1}{2}(x-1).\) If \((f(x))^{115}\) leaves a remainder of \(ax+b\) upon division by \(f\left(x^2\right),\) where \(a\) and \(b\) are constants, what is the value of \(9a+b?\)

If the remainder that polynomial \(x^3+6x^2+kx-1\) leaves upon division by \(x-1\) is \(766\), what is the value of constant \(k\)?

\( f(x) \) is a degree 4 polynomial satisfying \( f(n) = \frac {1} {n} \) for integers \( n = \) 1 to 5. If \( f(0) = \frac {a} {b} \), where \( a\) and \(b\) are coprime positive integers. What is the value of \( a + b\)?

Find the sum of the values among \( 3, 6, 9, 12\) that are roots of the polynomial \[2x^3-27x^2+63x+162.\]

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