Question

Find all the zeros of 2 x power 4 - 3 x cube minus 3 x square + 6 x + 2 if two zeros are under root 2 and minus under root 2​

Answer 1

AnswEr :

$\bf{\Large{\underline{\sf{Given\::}}}}$

The polynomials are $\sf{2x^{4} -3x^{3} -3x^{2} +6x-2}$ , if two zeroes are √2 and -√2.

$\bf{\Large{\underline{\sf{To\:find\::}}}}$

All the zeroes.

$\bf{\Large{\underline{\rm{\red{Explanation\::}}}}}$

We have;

$\leadsto\sf{\sqrt{2} \:\:\:\:\&\:\:\:-\sqrt{2} }$

So,

$\implies\tt{(x-\sqrt{2} )(x+\sqrt{2}) =0}\\\\\\\implies\tt{(x)^{2} -(\sqrt{2} )^{2} =0}\\\\\\\implies\tt{\green{x^{2} -2=0}}$

∴ x² - 2 is a factor of given polynomial.

$\begin{array}{r|lllll}& 2x^2 & -3x & +1 \\\cline{2-6} x^2 - 2 & 2x^4 & -3x^3 & -3x^2 & +6x & -2 \\ & 2x^4 & & -4x^2 \\ & - & & + \\ \cline{2-6} & & -3x^3 & +x^2 & +6x & -2 \\ & & -3x^3 & & +6x \\ & & + & & - \\ \cline{2-6} &&& +x^2 & & -2 \\ &&& +x^2 && -2 \\ &&& - && + \\ \cline{2-6} &&&& 00\end{array}$

Now,

We factorize 2x² - 3x + 1;

$\Rightarrow\tt{2x^{2} -3x+1=0}\\\\\\\Rightarrow\tt{2x^{2} -2x-x+1=0}\\\\\\\Rightarrow\tt{2x(x-1)-1(x-1)=0}\\\\\\\Rightarrow\tt{(x-1)(2x-1)=0}\\\\\\\Rightarrow\tt{x-1=0\:\:\:\:\:\:\:Or\:\:\:\:\:\:2x-1=0}\\\\\\\Rightarrow\tt{x=1\:\:\:\:\:Or\:\:\:\:\:2x=1}\\\\\\\Rightarrow\tt{\orange{x=1\:\:\:\:\:Or\:\:\:\:\:\:x=\dfrac{1}{2} }}$

∴ The zeroes of p(x) are √2, -√2, 1/2 & 1.

Answer 2

Step-by-step explanation:

Hope it's helpfull my friends