Question

What must be subtracted from x ^ 4 + 2x ^ 3 - 2x ^ 2 + 4x + 5 so that the result is exactly divisible by x ^ 2 + 2x - 3​

Answer 1

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Answer 2

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

Here, Dividend is

$\rm \: f(x) = {x}^{4} + {2x}^{3} - {2x}^{2} + 4x + 5$

and

Divisor is

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

We know,

Dividend = Divisor × Quotient + Remainder

$\rm\implies \:$Dividend - Remainder = Divisor × Quotient

It means, if we subtract Remainder from the Dividend, then it will be exactly divisible by the Divisor.

So, using Long Division, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: {x}^{2} + 1\:\:}}}\\ {\underline{\sf{ {x}^{2} + 2x - 3}}}& {\sf{\: {x}^{4} + 2{x}^{3} - {2x}^{2} + 4x + 5\:\:}} \\{\sf{}}& \underline{\sf{ \: \: \: \: \: \: \:- {x}^{4} - 2{x}^{3} + {3x}^{2} \: \: \: \: \: \: \: \: \: \:    \:  \:  \:  \:  \:  \:  \: \:\:}} \\ {{\sf{}}}& {\sf{\: \: \: \: \: \: \: \: \: {x}^{2} + 4x + 5 \: }} \\{\sf{}}& \underline{\sf{ \: \: \: \: \: \: \: \: \: - {x}^{2} - 2x + 3 \: \:   \:\:}} \\ {\underline{\sf{}}}& {\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2x + 8\:\:}}  \end{array}\end{gathered}\end{gathered}\end{gathered}$

So, from this long division we have,

$\rm \: Remainder \: = \: 2x + 8$

So,

$\rm\implies \:2x + 8 \: must \: be \: subtracted \: from \: \\ \rm \: \: \: \: \: \: \: \: \: \: {x}^{4} + {2x}^{3} - {2x}^{2} + 4x + 5 \: so \: that \: \\ \rm \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: it \: is \: exactlydivisible \: by \: {x}^{2} + 2x - 3.$

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Additional Information :-

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