Question

Find the other zeroes of the polynomial x^4_5x^3+2x^2+10x_8 if it is given that two of its zeroes are _root 2 and root 2​

Answer 1

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

$\rm :\longmapsto\: Let \: f(x) \: = \:{x}^{4}-5 {x}^{3}+{2x}^{2} +10x-8$

Now,

Given that

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

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

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

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

So,

Using long division method,

We have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: {x}^{2} - 5x + 4\:\:}}}\\ {\underline{\sf{ {x}^{2} - 2}}}& {\sf{\: {x}^{4} - 5{x}^{3} + {2x}^{2} + 10x - 8 \:\:}} \\{\sf{}}& \underline{\sf{ \: \: \: \: \: \: \: \: \: - {x}^{4} \: \: \: \: \: \: \: \: \: + 2 {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }} \\ {{\sf{}}}& {\sf{\: \: \:  \:  \:  \: \: -  5{x}^{3} + {4x}^{2} + 10x - 8 \:  \:  \:  \:   \:  \:  \:  \:\:}} \\{\sf{}}& \underline{\sf{\:\: \:  \: \: \: {5x}^{3}  \: \: \: \: \: \: \: \: \: - 10x  \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \: \: \: \: \: \: \: \: 4 {x}^{2} \: \: \: \: \: \: - 8 \:\:}} \\{\sf{}}& \underline{\sf{\: \:  \:\:  \: \: \: \: \: \: \: \: \: \: - {4x}^{2} \: \: \: \: \: + 8 \: \: \: \: \: \: \: \: \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \:  \:  \:  \:  \:  \: 0\:\:}}  \end{array}\end{gathered}\end{gathered}\end{gathered}$

We know,

Dividend = Divisor × Quotient + Remainder

$\rm :\longmapsto\: \: \:{x}^{4}-5 {x}^{3}+{2x}^{2} +10x-8$

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

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

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

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

Hence,

• Remaining zeroes are 4 and 1.

Additional Information :-

Quadratic Polynomial

$\rm :\longmapsto\: \alpha,\beta \: are \: zeroes \: of \: {ax}^{2} + bx + c \: then \:$

$\green{ \boxed{ \bf \: \alpha + \beta = - \frac{b}{a}}}$

$\green{ \boxed{ \bf \: \alpha \beta = \frac{c}{a}}}$

Cubic Polynomial

$\rm :\longmapsto\: \alpha,\beta, \gamma \: are \: zeroes \: of \: {ax}^{3} + {bx}^{2} + cx + d \: then \:$

$\green{ \boxed{ \bf \: \alpha + \beta + \gamma = - \frac{b}{a}}}$

$\green{ \boxed{ \bf \: \alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}}}$

$\green{ \boxed{ \bf \: \alpha\beta \gamma = - \frac{d}{a}}}$