Question

Obtain other zeros of the polynomial
$\tt{f(x) = {2x}^{4} + {3x}^{3} - 5 {x}^{2} - 9x - 3 }$
if two of it's zeros are
$\tt{ \sqrt{3} \: and \: - \sqrt{3} }$

​

Answer 1

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

Given that

$\tt :\longmapsto\:f(x) = {2x}^{4} + {3x}^{3} - {5x}^{2} - 9x - 3$

and

$\rm :\longmapsto\: - \sqrt{3} \: and \: \sqrt{3} \: are \: zeroes \: of \: f(x)$

$\rm :\longmapsto\:(x - \sqrt{3} )\: and \: (x + \sqrt{3}) \: are \: factors \: of \: f(x)$

$\rm :\longmapsto\:(x - \sqrt{3} )(x + \sqrt{3}) \: is \: factor \: of \: f(x)$

$\rm :\longmapsto\: {x}^{2} - 3 \: is \: factor \: of \: f(x)$

Using Long Division Method, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\: \: \: \: \: \: \: {2x}^{2} \: - \: 3x \: + 1 \: \: \: \: \:\:}}}\\ {{\sf{ {x}^{2} - 3}}}& {\sf{\: {2x}^{4} + {3x}^{3} - {5x}^{2} - 9x - 3 \:}} \\{\sf{}}&\underline{\sf{\: \: \: \: \: { - 2x}^{4} \: \: \: \: \: \: \: \: + 6 {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}}\\{\sf{}}&{\sf{\: \: \: \: \ \: {3x}^{3} + {x}^{2} - 9x - 3\:\:}}\\{\sf{}}&\underline{\sf{- {3x}^{3} \: \: \: \: \: \: \: \: + 9x \:\:}}\\{\sf{}}&{\sf{\: \: \: \: \ \: {x}^{2} - 3 \:\:}}\\{\sf{}}&\underline{\sf{\:\: \: \: \: - {x}^{2} + 3\:\:}}\\{\sf{}}&\underline{\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 0\:\: \: \: \: \: \: \: \: \: }}\end{array}\end{gathered}\end{gathered}\end{gathered}$

So, we get

$\rm :\longmapsto\:Dividend = {2x}^{4} + {3x}^{3} - 5 {x}^{2} - 9x - 3$

$\rm :\longmapsto\:Quotient = {2x}^{2} - 3x + 1$

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

$\rm :\longmapsto\:Remainder = 0$

We know that,

$\rm :\longmapsto\:Dividend = Divisor \times Quotient + Remainder$

So,

$\rm :\longmapsto\:{2x}^{4} + {3x}^{3} - 5 {x}^{2} - 9x - 3 = ( {x}^{2} - 3)( {2x}^{2} - 3x + 1)$

$\rm \: = \: \: ( {x}^{2} - 3)( {2x}^{2} - 2x - x + 1)$

$\rm \: = \: \: ( {x}^{2} - 3)\bigg(2x(x - 1) - 1(x - 1) \bigg)$

$\rm \: = \: \: ( {x}^{2} - 3)(x - 1)(2x - 1)$

Hence,

Other Zeroes of

$\tt :\longmapsto\:f(x) = {2x}^{4} + {3x}^{3} - {5x}^{2} - 9x - 3$

$\rm :\longmapsto\:are \: 1 \: and \: \dfrac{1}{2}$