Gujarat BoardEnglish MediumSTD 9MathsPolynomials3 Marks
Question
Factorise: $2x^3 - 3x^2 - 17x + 30$
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Answer
Let $p ( x )=2 x ^3-3 x ^2-17 x +30$ Constant term of $p ( x )=30$
$\therefore$ Factors of $30$ are $\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$ By trial,
we find that $p(2)=0$, so $(x-2)$ is a factor of $p(x)$.
$\left[\therefore 2(2)^3-3(2)^2-17(2)+30=16-12-34+30=0\right]$
Now, we see that $2 x^3-3 x^2-17 x+30=2 x^3-4 x^2+x^2-2 x-15 x+30$
$=2 x^2(x-2)+x(x-2)-15(x-2)$
$=(x-2)\left(2 x^2+x-15\right)[\text { taking }(x-2) \text { common factor }]$
Now, $\left(2 x^2+x-15\right)$ can be factorised either by spliting the middle term or by using the factor theorem.
Now, $\left(2 x^2+x-15\right)=2 x^2+6 x-5 x-15=2 x(x+3)-5(x+3)$ [by spliting the middle term]
$=(x+3)(2 x-5)$
$\therefore 2 x^3-3 x^2-17 x+30=(x-2)(x+3)(2 x-5)$
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