Factor $x^4-81$
A) $(x+3) 2(x-3) 2$
B) $\left(x^2+9\right)(x+3)(x-3)$
C) $\left(x^2+3\right)\left(x^2-3\right)$
D) $\left(x^2-9\right)(x+3)(x-3)$
Answer (2)
We factor the expression x 4 − 81 using the difference of squares method.
Recognize x 4 − 81 as a difference of squares: ( x 2 ) 2 − 9 2 .
Apply the difference of squares factorization: ( x 2 − 9 ) ( x 2 + 9 ) .
Recognize x 2 − 9 as another difference of squares: ( x − 3 ) ( x + 3 ) .
The completely factored form is ( x 2 + 9 ) ( x + 3 ) ( x − 3 ) , so the answer is ( x 2 + 9 ) ( x + 3 ) ( x − 3 ) .
Explanation
Understanding the Problem We are given the expression x 4 − 81 to factor. Our goal is to rewrite it as a product of simpler expressions.
Recognizing Difference of Squares Notice that x 4 can be written as ( x 2 ) 2 and 81 can be written as 9 2 . Thus, we have a difference of squares: x 4 − 81 = ( x 2 ) 2 − 9 2 .
Applying Difference of Squares Recall the difference of squares factorization: a 2 − b 2 = ( a − b ) ( a + b ) . Applying this to our expression, we get: ( x 2 ) 2 − 9 2 = ( x 2 − 9 ) ( x 2 + 9 ) .
Recognizing Another Difference of Squares Now, observe that x 2 − 9 is also a difference of squares, since 9 = 3 2 . So, x 2 − 9 = x 2 − 3 2 .
Applying Difference of Squares Again Applying the difference of squares factorization again, we have: x 2 − 9 = ( x − 3 ) ( x + 3 ) .
Combining the Factors Substituting this back into our expression, we get: ( x 2 − 9 ) ( x 2 + 9 ) = ( x − 3 ) ( x + 3 ) ( x 2 + 9 ) .
Final Answer So, the completely factored form of x 4 − 81 is ( x 2 + 9 ) ( x + 3 ) ( x − 3 ) . Comparing this with the given options, we see that option B matches our result.
Examples
Factoring polynomials like x 4 − 81 is a fundamental skill in algebra and is used in many areas of mathematics and engineering. For example, when designing a bridge, engineers use polynomial equations to model the forces and stresses acting on the structure. Factoring these polynomials helps them find critical points and ensure the bridge's stability. Similarly, in physics, factoring polynomials can help solve equations related to motion and energy. Understanding how to factor complex expressions allows for simplification and easier manipulation of equations in various real-world applications.
To factor x 4 − 81 , recognize it as a difference of squares, resulting in ( x 2 − 9 ) ( x 2 + 9 ) . Further factoring gives ( x − 3 ) ( x + 3 ) ( x 2 + 9 ) . The answer that matches this result is option B: ( x 2 + 9 ) ( x + 3 ) ( x − 3 ) .
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