Question

5. (i) The product of two expressions is x^5 +x^3 +x. If one of the two expressions is x^2+x+1, find the other expression.
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Answer 1

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

Given that,

$\rm \: First\:expression = {x}^{2} + x + 1$

and

$\rm \: Product\:of\:two\:expressions = {x}^{5} + {x}^{3} + x$

$\rm \: = \: x( {x}^{4} + {x}^{2} + 1)$

can be rewritten as

$\rm \: = \: x( {x}^{4} + {x}^{2} + {x}^{2} - {x}^{2} + 1)$

$\rm \: = \: x( {x}^{4} + 2{x}^{2} - {x}^{2} + 1)$

can be re-arranged as

$\rm \: = \: x( {x}^{4} + 2{x}^{2} + 1 - {x}^{2})$

$\rm \: = \: x\bigg[ \bigg({( {x}^{2}) }^{2} + 2 \times {x}^{2} \times 1 + {1}^{2}\bigg) - {x}^{2}\bigg]$

We know,

$\boxed{ \rm{ \: {x}^{2} + 2xy + {y}^{2} = {(x + y)}^{2} \: }}$

So, using this result, we get

$\rm \: = \: x\bigg[ {( {x}^{2} + 1)}^{2} - {x}^{2}\bigg]$

We know,

$\boxed{ \rm{ \: {x}^{2} - {y}^{2} = (x + y)(x - y) \: }}$

So, using this result, we get

$\rm \: = \: x\bigg[( {x}^{2} + x + 1)( {x}^{2} + 1 - x) \bigg]$

So,

$\rm\implies \:Product\:of\:two\:expressions = x( {x}^{2} + 1 + x)( {x}^{2} + 1 - x)$

Now,

$\rm \: Other\:expression = \dfrac{Product\:of\:two\:expressions}{First\:expression}$

$\rm \: = \: \dfrac{x( {x}^{2} + 1 + x)( {x}^{2} + 1 - x)}{ {x}^{2} + x + 1}$

$\rm \: = \: x({x}^{2} + 1 - x)$

$\rm \: = \: {x}^{3} + x - {x}^{2}$

$\rm \: = \: {x}^{3} - {x}^{2} + x$

Hence,

$\rm\implies \:\boxed{ \rm{ \:Other\:expression = \: {x}^{3} - {x}^{2} + x \: }}$

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

Additional Information :-

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