Question

How do you factor $f(x) = {x^4} - 9{x^2}?$

Answer 1

Hint:Look for if you can fit the given expression in an algebraic identity and then factorize it further.The following algebraic identity will be used to factorize the given expression: ${a^2} - {b^2} = (a + b)(a - b)$

Complete step by step answer:
In order to factorize the given expression, we have to see the expression that if it can fit into an algebraic identity or not. So let us see the expression
$f(x) = {x^4} - 9{x^2}$
With the help of law of indices for brackets, we can write ${x^4} = {\left( {{x^2}} \right)^2}$
$f(x) = {\left( {{x^2}} \right)^2} - 9{x^2}$
And also we know that nine is equals to three raise to the two, i.e. $9 = {3^2}$
So we can write the expression further as
$
f(x) = {\left( {{x^2}} \right)^2} - 9{x^2} \\
\Rightarrow f(x) = {\left( {{x^2}} \right)^2} - {3^2}{x^2} \\ $
Now we know that if two different bases which are being multiplied, have the same exponent or power then they can be written as their product (product of the bases) raise to the power.
With the help of this we can further write it as
$
f(x) = {\left( {{x^2}} \right)^2} - {3^2}{x^2} \\
\Rightarrow f(x) = {\left( {{x^2}} \right)^2} - {(3x)^2} \\ $
Now look at the expression, it is pretty much similar to the following algebraic identity
${a^2} - {b^2} = (a + b)(a - b)\;{\text{where}}\;a = {x^2}\;{\text{and}}\;b = 3x$
Applying this identity we will get,
$
f(x) = {\left( {{x^2}} \right)^2} - {(3x)^2} \\
\Rightarrow f(x) = ({x^2} + 3x)({x^2} - 3x) \\ $
We can again see that in the factored expression $f(x) = ({x^2} + 3x)({x^2} - 3x)$ we can take $x$ as common factor
$
f(x) = ({x^2} + 3x)({x^2} - 3x) \\
\Rightarrow f(x) = \left\{ {x(x + 3)} \right\}\left\{ {x({x^2} - 3x)} \right\} \\
\therefore f(x) = {x^2}(x + 3)(x - 3) \\
 $
So $f(x) = {x^2}(x + 3)(x - 3)$ is the factored form of the expression $f(x) = {x^4} - 9{x^2}$.

Note:On factoring these types of expressions, always try to take out the common factors from all the terms (if exists). Then further factorize the expression, because this will make the expression look more simplified than earlier.Algebraic identities are very useful in terms of factoring any expression and also evaluating any term so learning them is anyways good for you.