Question

Factorise each of the following:
(i) $27y^3+125z^3$
(ii) $64m^3-343n^3$

Answer 1

(i) $27y^3+125z^3$

=${\left(3y\right)}^3+{\left(5z\right)}^3$

We have an identity: $a^3+b^3=(a+b)(a^2-ab+b^2)$ (1)

In this problem, we have a=3y and b=5z, putting these values in (1), we get

${\left(3y\right)}^3+{\left(5z\right)}^3=(3y+5z)[{\left(3y\right)}^2-(3y\left)\right(5z)+{\left(5z\right)}^2)=(3y+5z)(9y^2-15yz+25z^2)$

(ii) $64m^3-343n^3$

=${\left(4m\right)}^3-{\left(7n\right)}^3$

We have an identity: $a^3-b^3=(a-b)(a^2+ab+b^2)$ (1)

In this problem, we have a=4m and b=7n, putting these values in (1), we get

${\left(4m\right)}^3-{\left(7n\right)}^3=(4m-7n)[{\left(4m\right)}^2+(4m\left)\right(7n)+{\left(7n\right)}^2)=(4m-7n)(16m^2+28mn+49n^2)$