Question

a quartic polynomial f(x) has 2 rational roots at (2,0) and (-1,0) and root at (-1±√3,0) if f(-2) is -2, then what is f(3) a-(-13/3) b-(13) c-(13/3) d--(-13) e-(answer is not there)

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

A quartic polynomial f(x) has 2 rational roots at (2,0) and (-1,0) and root at ( - 1 ± √3,0 ) if f(-2) is -2, then f(3)

$\displaystyle\sf{a. \: \: - \frac{13}{3} }$

$\displaystyle\sf{b. \: \: 13}$

$\displaystyle\sf{c. \: \: \frac{13}{3} }$

$\displaystyle\sf{d. \: \: - 13}$

e. Answer is not there

EVALUATION

Here it is given that the quartic polynomial f(x) has 2 rational roots at (2,0) and (-1,0) and root at ( - 1 ± √3,0 )

So all the four roots are

2 , - 1 , - 1 + √3 , - 1 - √3

So the factors are

x - 2 , x + 1 , x + 1 - √3 , x + 1 + √3

So quartic polynomial f(x) is given by

$\sf{f(x) =A (x - 2)(x + 1)(x + 1 + \sqrt{3})(x + 1 - \sqrt{3}) }$

Where A is non zero constant to be determined

$\sf{ \implies \: f(x) =A (x - 2)(x + 1)[ {(x + 1)}^{2} - 3] }$

Now it is given that f(-2) = - 2

Which gives

$\sf{ \implies \: f( - 2) =A ( - 2 - 2)( - 2 + 1)[ {( - 2 + 1)}^{2} - 3] }$

$\sf{ \implies \: A \times ( - 4) \times ( - 1) \times ( - 2) = - 2 }$

$\displaystyle \sf{ \implies \: A = \frac{1}{4} }$

So the quartic polynomial f(x) is

$\displaystyle \sf{ f(x) = \frac{1}{4} (x - 2)(x + 1)[ {(x + 1)}^{2} - 3]}$

Now

$\displaystyle \sf{ f(3) = \frac{1}{4} (3- 2)(3 + 1)[ {(3 + 1)}^{2} - 3]}$

$\displaystyle \sf{ \implies \: f(3) = \frac{1}{4} \times 1 \times 4 \times [ {4}^{2} - 3]}$

$\displaystyle \sf{ \implies \: f(3) = \frac{1}{4} \times 1 \times 4 \times 13}$

$\displaystyle \sf{ \implies \: f(3) = 13}$

FINAL ANSWER

Hence the correct option is b. 13

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Answer 2

$\huge \tt answer \: : \: \boxed{ \tt b) \: 13}$