Question

How do you factor ${x^3} - 25x?$

Answer 1

Hint:First check if there any common factor between the terms of the given expression, if yes then take the common factor out and rewrite the expression after taking the common factor and check if there any algebraic identity could be applied or not, if yes then apply that. The algebraic identity ${a^2} - {b^2} = (a + b)(a - b)$ will be used to solve this question.

Complete step by step solution:
In order to factorize the expression ${x^3} - 25x$, we need first to check for any common factor between the terms of the given expression,

Let us check the common factor between ${x^3}$ and $25x$ by listing their factors as follows

$
{x^3} = x \times x \times x \\
25x = 5 \times 5 \times x \\
$
We can see that $x$ is the common factor between ${x^3}$ and $25x$
So taking out $x$ common in the expression ${x^3} - 25x$, we will get
$
= {x^3} - 25x \\
= x({x^2} - 25) \\
$
We got the new expression $ = x({x^2} - 25)$, but this is not the factored form of ${x^3} - 25x$
Check the expression $x({x^2} - 25)$, can we factorize it further?
Yes, we can factorize it furthermore by use of an algebraic identity which could be given as
${a^2} - {b^2} = (a + b)(a - b)$

See the $({x^2} - 25)$ in the expression $x({x^2} - 25)$, we can see that $({x^2} - 25)$ is
suitable for the above algebraic identity, so factoring further with the help of ${a^2} - {b^2} = (a + b)(a - b)$, we will get $ = x({x^2} - 25)$

We can write $25 = {5^2}$
$
= x({x^2} - {5^2}) \\
= x((x + 5)(x - 5)) \\
= x(x + 5)(x - 5) \\
$
Therefore $x(x + 5)(x - 5)$ is the required factored form of ${x^3} - 25x$

Note: In order to factorize any polynomial expression, checking for common factors is very helpful way that is if you get any common factor then after taking out the common factor the expression becomes more simplified and then for further factoring, other methods will be easily visible to you after taking out the common factor.