question_answer

Factorise and divide the following:

(a)\[\left( {{x}^{2}}-22x+117 \right)\div \left( x-13 \right)~~~~\]

(b) \[\left( 9{{x}^{2}}-4 \right)\div \left( 3x+2 \right)\]

Answer:

(a)  \[({{x}^{2}}-22x+117)\div (x-13)\]

\[\because \]\[{{x}^{2}}-22x+117={{x}^{2}}-(13+9)x+117\]

\[={{x}^{2}}-13x-9x+117\]

\[=x(x-13)-9(x-13)\]

\[=(x-13)(x-9)\]

\[\therefore \] \[\frac{{{x}^{2}}-22x+117}{(x-13)}=\frac{(x-13)(x-9)}{(x-13)}=x-9\]

(b) \[~\left( 9{{x}^{2}}-4 \right)\div \left( 3x+2 \right)\]

\[\because \]       \[9{{x}^{2}}-4={{(3x)}^{2}}-{{(2)}^{2}}\]

\[=\left( 3x+2 \right)\left( 3x-2 \right)\]

\[\therefore \]  \[\frac{9{{x}^{2}}-4}{(3x+2)}=\frac{(3x+2)(3x-2)}{(3x+2)}=(3x-2)\]