Question
Factorise: $\text{a}^3-\frac{1}{\text{a}^3}-2\text{a}+\frac{2}{\text{a}}$
✓
Answer
We know that, Since $a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
$\text{a}^3-\frac{1}{\text{a}^3}-2\text{a}+\frac{2}{\text{a}}$
$=\text{a}^3-\frac{1}{\text{a}^3}-2\Big(\text{a}-\frac{1}{\text{a}}\Big)$
$=\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}^2+\text{a}\times\frac{1}{\text{a}}+\frac{1}{\text{a}^2}\Big)-2\Big(\text{a}-\frac{1}{\text{a}}\Big)$
$=\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}^2+1+\frac{1}{\text{a}^2}-2\Big)$
$=\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}^2+\frac{1}{\text{a}^2}-1\Big)$
$\text{a}^3-\frac{1}{\text{a}^3}-2\text{a}+\frac{2}{\text{a}}$
$=\text{a}^3-\frac{1}{\text{a}^3}-2\Big(\text{a}-\frac{1}{\text{a}}\Big)$
$=\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}^2+\text{a}\times\frac{1}{\text{a}}+\frac{1}{\text{a}^2}\Big)-2\Big(\text{a}-\frac{1}{\text{a}}\Big)$
$=\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}^2+1+\frac{1}{\text{a}^2}-2\Big)$
$=\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}^2+\frac{1}{\text{a}^2}-1\Big)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Simplify the following products: $(\text{x}^2 +\text{x}- 2)(\text{x}^2-\text{x} + 2)$
→Diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect each other at $P$. Show that: $\text{ar}(\triangle\text{APB})\times\text{ar}(\triangle\text{CPD})\\=\text{ar}(\triangle\text{APD})\times\text{ar}(\triangle\text{BPC})$
→Find the following products:
$(3 x-4 y+5 z)\left(9 x^2+16 y^2+25 z^2+12 x y-15 z x+20 y z\right)$
→$(3 x-4 y+5 z)\left(9 x^2+16 y^2+25 z^2+12 x y-15 z x+20 y z\right)$
If $O$ is the centre of the circle, find the value of $x$ in the following figures:
→Simplify: $\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}$
→The following table shows the number of scooters sold by a dealer during six consecutive years. Draw a bar graph to represent this data.
→|
Year
|
$2011$
|
$2012$
|
$2013$
|
$2014$
|
$2015$
|
$2016$
|
|
Number of scooters sold (in thousands)
|
$16$
|
$20$
|
$32$
|
$36$
|
$40$
|
$48$
|
The measurements (in mm) of the diameters of the heads of $50$ screws are given below:
Calculate the mean diameter of the heads of the screws.
→|
Diameter (in mm)($x_i$)
|
$34$
|
$37$
|
$40$
|
$43$
|
$46$
|
|
Number of screws ($f_i$)
|
$5$ |
$10$
|
$17$
|
$12$
|
$6$
|
The capacity of a cuboidal tank is $50000$ litres of water. Find the breadth of the tank if its length and depth are respectively10m and $2.5\ m.$ (Given, $1000$ litres $= 1m^3.)$
→Show that: $2x - 3$ is a factor of $x + 2x^3 - 9x^2 + 12.$
→How many litres of water flows out of a pipe having an area of cross section of $5 cm^2$ in $1$ minute, if the speed of water in the pipe is $30 cm / sec$ ?
→