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Can someone define Synthetic division and whats its formula or method to solve a polynomial, for example:

1) Use synthetic division to find the value of $k$ if $-2$ is a zero of polynomial: $$x^{3} + 4x^{2} + kx + 8$$

2) Use synthetic division to find the value of $p$ and $q$ if $x+1$ and $x-2$ are the factors of $$x^{3}+ px^{2} + qx + 6$$

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Mechanically it is the same thing as polynomial long division. Unfortunately it is difficult to do the formatting here.

$x^3 + 4x^2 + kx + 8\\ (x+2)(x^2) - 2x^2 + 4x^2 + kx + 8\\ (x+2)(x^2) + 2x^2 + kx + 8\\ (x+2)(x^2 + 2x) - 4x + kx + 8\\ (x+2)(x^2 + 2x - 4+k) +8-2k + 8\\ (x+2)(x^2 + 2x - 4+k) -2k+16 \\ $

In a clumsy way I have divided the polynomial by $(x+2)$

If $-2$ is a root of the polynomial.

$(-2)^3 + 4(-2)^2 + k(-2) + 8 = 0$

But since we did that polynomial division. We don't need to plug $-2$ into every $x$ term to see that we can $0$ out a large block of the polynomial.

$(-2+2)(x^2 + 2x - 4+k) -2k+16 = 0$

$-2k+16 = 0\\k = 8$

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    Thank you Doug, I have done the synthetic division and then applied the last part of your answer and found the value of k = 8.. and i have already solved the part 2 of question.. – 2017-02-03