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Obtain all the zeros of the polynomial x^4+4x^3-2x^2-20x-15 if two of its zeros are root 5 and minus root 5.

Posted by Shraddha Jaiswal 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

Here f(x)=x⁴ + 4x³ - 2x² - 20x - 15

{tex}\sqrt { 5 }{/tex} and - {tex}\sqrt { 5 }{/tex} are zeros of f(x)
{tex}\begin{array}{l}\text{so (x-}\sqrt5)(\mathrm x+\sqrt{5)}=\mathrm x^2-5\;\mathrm{is}\;\mathrm a\;\mathrm{factor}\;\mathrm{of}\;\mathrm f(\mathrm x)\\\end{array}{/tex}
Dividing x⁴ + 4x³ - 2x² - 20x - 15 by x² - 5, we get

The quotient q(x)=x²+4x+3

=x²+x+3x+3

=x(x+1)+3(x+1)

=(x+3)(x+1)

q(x)=0 if x+3=0 or x+1 =0

Hence zeros of q(x) are -3 and -1
Thus, the zeros of the given polynomial f(x) are
{tex}\sqrt { 5 } , - \sqrt { 5 }{/tex} , -3 , -1

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