To find an equivalent equation to [tex]\( f(x) = 16x^4 - 81 = 0 \)[/tex], we will factor the given polynomial. Here is the step-by-step process:
1. Recognize the difference of squares: The given equation [tex]\( 16x^4 - 81 = 0 \)[/tex] is a difference of squares. We can express it as:
[tex]\[
(4x^2)^2 - 9^2 = 0
\][/tex]
2. Apply the difference of squares formula: The difference of squares formula states that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex]. For our expression, [tex]\( a = 4x^2 \)[/tex] and [tex]\( b = 9 \)[/tex]. Using this formula, we factor the equation:
[tex]\[
16x^4 - 81 = (4x^2 - 9)(4x^2 + 9)
\][/tex]
3. Further factor [tex]\( 4x^2 - 9 \)[/tex]: Notice that [tex]\( 4x^2 - 9 \)[/tex] is itself a difference of squares. We can factor it further using the same formula with [tex]\( a = 2x \)[/tex] and [tex]\( b = 3 \)[/tex]:
[tex]\[
4x^2 - 9 = (2x - 3)(2x + 3)
\][/tex]
4. Combine all factors: Substitute back the factors into the original expression we found in step 2:
[tex]\[
16x^4 - 81 = (2x - 3)(2x + 3)(4x^2 + 9)
\][/tex]
Therefore, the factored form of the equation [tex]\( 16x^4 - 81 = 0 \)[/tex] is:
[tex]\[
(2x - 3)(2x + 3)(4x^2 + 9) = 0
\][/tex]
This is the equivalent equation.
