Question 2: Factorize each of the following expresssions:
\((ii)\) \(\;\) \((x+2)(x-7)(x-4)(x-1)+17\)
Solution:
$\displaystyle\ \ \ \ ( x+2)( x-7)( x-4)( x-1) +17$
$\displaystyle =[( x+2)( x-7)][( x-4)( x-1)] +17$
$\displaystyle =[ x( x-7) +2( x-7)][ x( x-1)\ -$$4( x-1)]\ +17$
$\displaystyle =[ x^{2} -7x+2x-14][ x^{2} -x-4x+4]\ +\ $$17$
$\displaystyle =[ x^{2} -5x-14][ x^{2} -5x+4] +17$
Let \(\; y=x^2-5x\), we have
$\displaystyle =[ y-14][ y+4] +17$
$\displaystyle =y( y+4) -14( y+4) +17$
$\displaystyle =y^{2} +4y-14y-56+17$
$\displaystyle =y^{2} -10y-39$
$\displaystyle =y^{2} -13y+3y-39$
$\displaystyle =y( y-13) +3( y-13)$
$\displaystyle =( y-13)( y+3)$
$\displaystyle =\left( x^{2} -5x-13\right)\left( x^{2} -5x+3\right)$