Remainder Factor Theorem

Remainder Factor Theorem


If f(x)=(x14)(x+1)(x2+2x+6)f(x)=(x-14)(x+1)(x^2+2x+6), what is the sum of all real values xx satisfying f(x)=0f(x)=0?

Consider the polynomial f(x)=12(x1).f(x)=\frac{1}{2}(x-1). If (f(x))115(f(x))^{115} leaves a remainder of ax+bax+b upon division by f(x2),f\left(x^2\right), where aa and bb are constants, what is the value of 9a+b?9a+b?

If the remainder that polynomial x3+6x2+kx1x^3+6x^2+kx-1 leaves upon division by x1x-1 is 766766, what is the value of constant kk?

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

Find the sum of the values among 3,6,9,12 3, 6, 9, 12 that are roots of the polynomial 2x327x2+63x+162.2x^3-27x^2+63x+162.


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