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جستجو

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The polynomial $p\left( x \right)$ is defined by

$p\left( x \right) = \alpha {x^3} - {x^2} + 4x - \alpha $

where a is $\alpha $ constant. It is given that $\left( {2x - 1} \right)$ is a factor of $p\left( x \right)$.

a) Find the value of $\alpha $ and hence factorise $p\left( x \right)$.

b) When $\alpha $ has the value found in part (a), express $\frac{{8x - 13}}{{p\left( x \right)}}$ in partial fractions.

پاسخ تشریحی :
نمایش پاسخ

a) Substitute $x = \frac{1}{2}$ and equate to zero

or divide by $\left( {2x - 1} \right)$, reach $\frac{\alpha }{2}{x^2} + kx + ...$ and equate remainder to zero

or by inspection reach $\frac{\alpha }{2}{x^2} + bx + c$ nd an equation in b/c

or by inspection reach $A{x^2} + Bx + \alpha $ and an equation in $A/B$

Obtain $\alpha  = 2$

Attempt to find quadratic factor by division or inspection or equivalent

Obtain $\left( {2x - 1} \right)\left( {{x^2} + 2} \right)$

b) State or imply form $\frac{A}{{2x - 1}} + \frac{{Bx + C}}{{{x^2} + 2}}$, following factors from part (a)

Use relevant method to find a constant

Obtain $A =  - 4$, following factors from part (a)

Obtain $B = 2$

Obtain $C = 5$

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