Question

For what value of k , is the polynomial f(x) =3x4-9x3+x2+15x+k completely divisible by 3x2-5

Answer 1

Answer:

$k = - 10$

Step-by-step explanation:

Given :-

$= > 3 {x}^{4} - 9 {x}^{3} + {x}^{2} + 15x + k \: is \: divisible \: by \: 3 {x}^{2} - 5$

To find k

find the value of x from

$3 {x}^{2} - 5$

we get ,

$= > 3 {x}^{2} - 5 = 0 \\ = > 3 {x}^{2} = 5 \\ = > {x}^{2} = \frac{5}{3 } \\ = > x = \sqrt{ \frac{5}{3} }$

Now put the value in the polynomial:-

$= > 3 { ( \sqrt{ \frac{5}{3} }) }^{4} - 9 {( \sqrt{ \frac{5}{3} } ) }^{3} + { (\sqrt{ \frac{5}{3} }) }^{2} + 15( \sqrt{ \frac{5}{3} } ) + k = 0 \\ = > 3 {( \frac{5}{3}) }^{2} - 9( \frac{5}{3} \sqrt{ \frac{5}{3} } ) + \frac{5}{3} + 15 \sqrt{ \frac{5}{3} } + k = 0 \\ = > \frac{75}{9} - \frac{45}{3} \frac{ \sqrt{5} }{ \sqrt{3} } + \frac{5}{3} + 15\sqrt{ \frac{5}{3} } + k = 0 \\ = > \frac{75 + 15}{9} - \frac{45 \sqrt{5} + 45 \sqrt{5} }{3 \sqrt{3} } + k = 0 \\ \\ = > - -\frac{90}{9} + 0 = k \\ k = - 10$

Hope it helps you.

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