Question

Using factor theorem, factorize each of the following polynomial:
x³-10x²-53x-42

Answer 1

$\textbf{Concept used:}$

$\textbf{Factor theorem:}$

$\text{(x-a) is a factor of f(x) iff f(a) =0}$

$\text{Let f(x)=}x^3-10x^2-53x-42$

$\text{put x=-1}$

$\text{f(-1)}=(-1)^3-10(-1)^2-53(-1)-42$

$\text{f(-1)}=-1-10+53-42=0$

$\therefore\textbf{(x+1) is a factor of f(x)}$

$\begin{array}{r|cccc}-1&1&-10&-53&-42\\&&-1&11&42\\\cline{2-5}&1&-11&-42&[0]\end{array}$

$\text{Now, we factorize }x^2-11x-42$

$=(x-14)(x+3)$

$\therefore\boxed{\bf\;x^3-10x^2-53x-42=(x+1)(x+3)(x-14)}$