Factorisation Using Algebraic Identities

A factor of $x^3-1$ is:

If $a^3+a=a\left(a^2+k\right)$ then find the value of $k.$

Factorise : $\left(x^6-64y^3\right)$

Factorize the expression ${\mathrm{c}}^3-27{\mathrm{b}}^3{\mathrm{a}}^3$ using ${\mathrm{a}}^3-{\mathrm{b}}^3=(\mathrm{a}-\mathrm{b})\left({\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorise the expression ${\mathrm{x}}^3{\mathrm{y}}^3-7^3$  using ${\mathrm{a}}^3-{\mathrm{b}}^3=(\mathrm{a}-\mathrm{b})\left({\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorize the expression $-16{\mathrm{y}}^3+2{\mathrm{x}}^3$ using ${\mathrm{a}}^3-{\mathrm{b}}^3=(\mathrm{a}-\mathrm{b})\left({\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorize the expression $3{\mathrm{b}}^3-192{\mathrm{c}}^3$ using ${\mathrm{a}}^3-{\mathrm{b}}^3=(\mathrm{a}-\mathrm{b})\left({\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorise the expression ${\mathrm{y}}^3-27$ using ${\mathrm{a}}^3-{\mathrm{b}}^3=(\mathrm{a}-\mathrm{b})\left({\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorize the expression $\left({\mathrm{r}}^4+27{\mathrm{p}}^3\mathrm{r}\right)$ using ${\mathrm{a}}^3+{\mathrm{b}}^3=(\mathrm{a}+\mathrm{b})\left({\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorise the expressions $64{\mathrm{m}}^3+{\mathrm{n}}^3$ using ${\mathrm{a}}^3+{\mathrm{b}}^3=(\mathrm{a}+\mathrm{b})\left({\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorize the expression $x^3{y}^3+27$ using ${\mathrm{a}}^3+{\mathrm{b}}^3=(\mathrm{a}+\mathrm{b})\left({\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Factorize expressions $2{\mathrm{a}}^3+16$  using ${\mathrm{a}}^3+{\mathrm{b}}^3=(\mathrm{a}+\mathrm{b})\left({\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2\right)$ identity.

Given are the sum and product of two numbers. Find the two numbers.

If the factors of $x^2-10x+25$ are in the form of ${\left(x-k\right)}^l$ then, find the value of $k+l$.

Factorise the following polynomial.

$p^2-7p-18$