What the number should be subtracted from ${2 x}^{3} - {5 x}^{2} + 5 x$ so that the resulting polynomial has $2 x - 3$ as a factor?

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Solution

Let the number which should be subtracted from the polynomial $2 x^{3} - 5 x^{2} + 5 x$ be $a .$

Let us say,
$p (x) = 2 x^{3} - 5 x^{2} + 5 x - a$

By using the factor theorem,
$2 x - 3 = 0$
$\Rightarrow x = \frac{3}{2}$

$\begin{matrix} p (\frac{3}{2}) & = 0 2 {(\frac{3}{2})}^{3} - 5 {(\frac{3}{2})}^{2} + 5 (\frac{3}{2}) - a & = 0 \frac{27}{4} - \frac{45}{4} + \frac{15}{2} - a & = 0 a & = \frac{15}{2} - \frac{9}{2} = 3 \end{matrix}$

Hence, $3$ should be subtracted from $2 x^{3} - 5 x^{2} + 5 x$ so, that result will have $2 x - 3$ as its factor.

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What the number should be subtracted from 2x3−5x2+5x so that the resulting polynomial has 2x−3 as a factor?

What the number should be subtracted from ${2 x}^{3} - {5 x}^{2} + 5 x$ so that the resulting polynomial has $2 x - 3$ as a factor?