Question

Find the factors of the polynomial given below (x^2 - 6x)^2 +12 (x^2 - 6x) +35
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Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: {( {x}^{2} - 6x)}^{2} + 12( {x}^{2} - 6x) + 35$

Let assume that

$\red{\rm \: {x}^{2} - 6x = y \: }$

So, above expression can be rewritten as

$\rm \: = \: {y}^{2} + 12y + 35$

$\rm \: = \: {y}^{2} + 7y + 5y + 35$

$\rm \: = \: y(y + 7) + 5(y + 7)$

$\rm \: = \: (y + 7)(y + 5)$

$\rm \: = \: ( {x}^{2} - 6x + 7)( {x}^{2} - 6x + 5)$

$\rm \: = \: ( {x}^{2} - 6x + 7)( {x}^{2} - 3x - 2x + 5)$

$\rm \: = \: ( {x}^{2} - 6x + 7)\bigg(x(x - 3) - 2(x - 3)\bigg)$

$\rm \: = \: ( {x}^{2} - 6x + 7)(x - 2)(x - 3)$

Hence,

$\rm\implies \: {( {x}^{2} - 6x) }^{2} + 12( {x}^{2} - 6x) + 35 \\ \\ \rm \: \red{ \: = \: \: ( {x}^{2} - 6x + 7)(x - 2)(x - 3)}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$