Question

determine the value of 'b' for which the polynomial 5x^3-x^2+4x+b is divisible by 1-5x

Answer 1

Please refer the above photograph for the used process
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Answer 2

$f(x) = {5x}^{3} - {x}^{2} + 4x - b \\ now \\ to \: find \: out \: the \: value \: of \: x \\ 1 - 5x = 0 \\ = \: \: - 5x = - 1 \\ = \: \: x = \frac{ - 1}{ - 5} \\ \\ = x = \frac{1}{5} \\ \\ f(x) = {5x}^{3} - {x}^{2} + 4x + b \\ f( \frac{1}{5} ) = 5( \frac{1}{5} )^{3} - ( \frac{1}{5} )^{2} + 4( \frac{1}{5} ) + b \\ \\ = 5 \times \frac{1}{125} - \frac{1}{25} + 4 \times \frac{1}{5} + b = 0 \\ \\ after \: solving \: we \: get \\ \frac{1}{25} - \frac{1}{25} + \frac{4}{5} + b = 0 \\ taking \: lcm \: we \: get \\ \frac{1 - 1 + 4 \times 5}{25} + b = 0 \\ \\ \frac{20}{25} + b = 0 \\ \\ = \frac{4}{5} + b = 0 \\ \\ = \: \: b = \frac{5}{4}$
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@Mahak24

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