Question

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

Answer 1

Hint: In this given problem, the given equation is a perfect square equation. Here we have to factor the given equation. We know that the given equation is in perfect square, so we can reduce it in an easier way to find the factors of the given equation. Here we have to know the algebraic formula for the difference of the square to find the factor of the equation. Here first we will get the factors by using the formula, then we can simply until we get the final factor.

Complete step by step answer:
We know that the given equation is,
\[16{{x}^{4}}-81\]…….. (1)
Now we can simplify this equation (1) to get
\[\Rightarrow {{\left( 4{{x}^{2}} \right)}^{2}}-{{9}^{2}}\]……. (2)
We also know that the difference of the square is,
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]…… (3)
Now we can apply the formula (3) in the equation (2), we get
\[\Rightarrow {{\left( 4{{x}^{2}} \right)}^{2}}-{{9}^{2}}=\left( 4{{x}^{2}}+9 \right)\left( 4{{x}^{2}}-9 \right)\]…… (4)
We know that from the above factors, we can write \[4{{x}^{2}}-9\] in the form of the formula (3), we get
\[\Rightarrow {{\left( 2x \right)}^{2}}-{{\left( 3 \right)}^{2}}=\left( 2x-3 \right)\left( 2x+3 \right)\]
Now we can write the above two factors in (4).
\[\Rightarrow \left( 4{{x}^{2}}+9 \right)\left( 2x+3 \right)\left( 2x-3 \right)\]

Therefore, the factors of \[16{{x}^{4}}-81\] are \[\left( 4{{x}^{2}}+9 \right)\left( 2x+3 \right)\left( 2x-3 \right)\].

Note: Students make mistakes in the algebraic formula part, always remember the algebraic formula, it is used in many problems. Since it is a perfect square problem, we can solve and find the factors by simplifying it and using formula methods, if it is not in complete square, then we have to solve it in a different method.