Question

Obtain all other zeros of (x4 + 4x3 - 2x2 - 20x - 15) if two of its zeros are √5 and -√5 .

Answer 1

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

Given polynomial is

$\rm :\longmapsto\: {x}^{4} + {4x}^{3} - {2x}^{2} - 20x - 15$

Let assume that

$\rm :\longmapsto\:f(x) = {x}^{4} + {4x}^{3} - {2x}^{2} - 20x - 15$

Now, Further given that

$\rm :\longmapsto\: \sqrt{5} \: and \: - \sqrt{5} \: are \: zeroes \: of \: f(x)$

$\rm :\longmapsto\: x - \sqrt{5} \: and \: x + \sqrt{5} \: are \: factors \: of \: f(x)$

$\rm :\longmapsto\: (x - \sqrt{5}) (x + \sqrt{5}) \: is \: factor \: of \: f(x)$

$\rm :\longmapsto\: ( {x}^{2} - {( \sqrt{5} )}^{2} ) \: is \: factor \: of \: f(x)$

$\rm :\longmapsto\: {x}^{2} - 5 \: is \: factor \: of \: f(x)$

So, Using Long Division, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\: \: \: \: {x}^{2} + 4x + 3 \: \: \:}}}\\ {{\sf{ {x}^{2} - 5}}}& {\sf{\: {x}^{4} + {4x}^{3} - {2x}^{2} - 20x - 15 \:}} \\{\sf{}}&\underline{\sf{\: \: \: { -x}^{4} + \: \: \: \: \: \: \: \: \: 5{x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}}\\{\sf{}}&{\sf{\: \: \: \: \: \: \: \: \: \: {4x}^{3} + {3x}^{2} - 20x - 15 \:\:}}\\{\sf{}}&\underline{\sf{\:\: \: \: \: \: \: \: \: \: - {4x}^{3} \: \: \: \: \: \: \: \: \: + 20x \: \: \: \: \: \: \: \: \: \:\:}}\\{\sf{}}&{\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {3x}^{2} \: \: \: \: \: \: \: \: \: \: - 15\:\:}}\\{\sf{}}&\underline{\sf{\:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { - 3x}^{2} \: \: \: \: \: \: \: \: \: + 15 \:\:}}\\{\sf{}}&\underline{\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 0 \: \: \: \: \: \: \: \: }}\end{array}\end{gathered}\end{gathered}\end{gathered}$

So, By Euclid Division Algorithm, we have

↝ Dividend = Divisor × Quotient + Remainder

$\rm :\longmapsto\: {x}^{4} + {4x}^{3} - {2x}^{2} - 20x - 15$

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

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

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

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

• So, Remaining zeroes are - 1 and - 3.