question_answer

If a polynomial \[{{x}^{4}}+5{{x}^{3}}+4{{x}^{2}}-10x-12\] has two zeroes as \[-2\] and \[-3\], then find the other zeroes.

Answer:

Given, polynomial is \[{{x}^{4}}+5{{x}^{3}}+4{{x}^{2}}-10x-12\].

Since two zeroes are \[-2\] and \[-3\]

\[\therefore (x+2)(x+3)={{x}^{2}}+3x+2x+6\]

Dividing the polynomial with\[{{x}^{2}}+5x+6\],

\[\therefore \,\,{{x}^{4}}+5{{x}^{3}}+4{{x}^{2}}-10x-12\]

\[=({{x}^{2}}+5x+6)({{x}^{2}}-2)\]

\[=(x+2)(x+3)(x-\sqrt{2})(x+\sqrt{2})\]

Other zeroes: \[x-\sqrt{2}=0\]   or \[x+\sqrt{2}=0\]

\[x=\sqrt{2}\]          or      \[x=-\sqrt{2}\]

The zeros of the polynomial are \[-2,-3,\sqrt{2}\] and \[-\sqrt{2}\]