Algebra

# 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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