Question

find the value of 'k' if (3x+2) is the factor of 4x^3+3x^2-4x+k​

Answer 1

Here is your answer hope it is helpful.

Answer 2

$\boxed{\sf{{p(x) = 4 {x}^{3} + 3 {x}^{2} - 4x + k}}}$

Also,

$\boxed{3x + 2}$

Is a factor of p(x)

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According to $\sf{REMAINDER\:THEOREM}$ :-

If x-a is a factor of p(x)

Then

P(a) will be equal to Zero.

So,

$3x + 2 = 0 \\ \\ 3x = - 2 \\ \\ x = \frac{ - 2}{3}$

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As per Remainder theorem :-

$\boxed{ \boxed{p( \frac{ - 2}{3}) = 0}}$

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$\boxed{4 {( \frac{ - 2}{3}) }^{3} + 3 {( \frac{ - 2}{3})}^{2} - 4( \frac{ - 2}{3}) + k = 0}\\ \\ \\ \boxed{4 \times \frac{( - 8)}{27} + 3 \times \frac{(4)}{9} + \frac{8}{3} + k = 0}\\ \\ \\ \boxed{\frac{ - 32}{27} + \frac{12}{9} + \frac{8}{3} + k = 0} \\ \\ \\ \boxed{\frac{ - 32 + 36 + 72}{27} + k = 0}\\ \\ \\ \boxed{k + \frac{76}{27} = 0}\\ \\ \\ \boxed{k = \frac{ - 76}{27}}$

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