Question

find all zeros of the polynomial x⁴ +x³ -14x² -2x +24 .if two of its zeroes are √2 and -√2 / by division method ​

Answer 1

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

Given polynomial is

$\rm :\longmapsto\:f(x) = {x}^{4} + {x}^{3} - {14x}^{2} - 2x + 24$

Further given that,

$\rm :\longmapsto\: - \sqrt{2} \: and \: \sqrt{2} \: are \: zeroes \: of \: f(x)$

$\rm :\longmapsto\:(x + \sqrt{2}) \: and \: (x - \sqrt{2}) \: are \: factors \: of \: f(x)$

$\rm :\longmapsto\:(x + \sqrt{2})(x - \sqrt{2}) \: is \: factors \: of \: f(x)$

$\rm :\longmapsto\: {x}^{2} - {( \sqrt{2})}^{2} \: is \: factors \: of \: f(x)$

$\rm :\longmapsto\: {x}^{2} - 2\: is \: factors \: of \: f(x)$

So, by using long division, we have

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: {x}^{2} + x - 2\:\:}}}\\ {\underline{\sf{ {x}^{2} - 2}}}& {\sf{\: {x}^{4} + {x}^{3} - {14x}^{2} - 2x + 24 \:\:}} \\{\sf{}}& \underline{\sf{-{x}^{4} \: \: \: \: \: \: \: \: + {2x}^{2}   \:  \:  \:  \:  \:  \:  \:  \:  \: \:\:}} \\ {{\sf{}}}& {\sf{\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:\: {x}^{3} - {12x}^{2} - 2x + 24  \:  \:  \:  \:   \:  \:  \:  \:\:}} \\{\sf{}}& \underline{\sf{\:\: \:  \:  \:  - {x}^{3}  \: \: \: \: \: \: \: \: + \: 2x  \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \:  \:  \:  \:  \ - {12x}^{2} + 24  \:\:}} \\{\sf{}}& \underline{\sf{\: \:  \:  \:  \:    \: + {12x}^{2} - 24\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \:  \:  \:  \:  \:  \:  \:  \:0\:\:}}  \end{array}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

Now, we know that,

Dividend = Divisor × Quotient + Remainder

So,

$\rm :\longmapsto\:{x}^{4} + {x}^{3} - {14x}^{2} - 2x + 24 = ( {x}^{2} - 2)( {x}^{2} + x - 2)$

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

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

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

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

Hence,

• Remaining zeroes are - 2 and 1.