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How do you factor \[y=16{{x}^{4}}-81\]?

Last updated date: 27th Feb 2024
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IVSAT 2024
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Hint: We will solve this problem by factoring method. First we will rewrite the equation to factor the given equation. We will derive the factors from the equation using the formula we have and then simplify them to arrive at the solution.
The formulas we use to solve the problem are
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]

Complete step by step solution:
Given equation is
First we have to rewrite the equation so that we can derive the formula.
By seeing the equation we can see that we can write \[16{{x}^{4}}\] as \[{{\left( 4{{x}^{2}} \right)}^{2}}\] and also we can write \[81\] as \[{{\left( 9 \right)}^{2}}\]. So by substituting the values we can rewrite the equation as
\[\Rightarrow {{\left( 4{{x}^{2}} \right)}^{2}}-{{\left( 9 \right)}^{2}}\]
By seeing the equation we can say that it is in the form of \[{{a}^{2}}-{{b}^{2}}\]. So we can apply the formula above discussed.
We will derive the equation using the formula
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
By applying the formula we get the equation as
\[\Rightarrow \left( 4{{x}^{2}}-9 \right)\left( 4{{x}^{2}}+9 \right)\]
Now again in the equation we can rewrite the first term into factors.
We can see that we can write \[4{{x}^{2}}\] as \[{{\left( 2x \right)}^{2}}\] and also we can write \[9\] as \[{{\left( 3 \right)}^{2}}\].
By substituting these values rewrite the first term as
\[\Rightarrow \left( 4{{x}^{2}}-9 \right)=\left( 2x+3 \right)\left( 2x-3 \right)\]
By rewriting the first term now the equation will look like
\[\Rightarrow \left( 2x+3 \right)\left( 2x-3 \right)\left( 4{{x}^{2}}+9 \right)\]
We cannot further simplify it because the second term doesn’t have real roots.
So that by solving the given equation we got the factors as \[\left( 2x+3 \right)\left( 2x-3 \right)\left( 4{{x}^{2}}+9 \right)\].

Note: If we want to further simplify the second term we can do it by imaginary roots. We can factor it as \[\left( 2x+3i \right)\left( 2x-3i \right)\] and then we can write the factors. We should know how to rewrite the expression to apply the formulas otherwise solving these types of questions will be difficult.