Factor $64 x^3-1$.
A. $(4 x+1)(16 x^2+4 x+1)$
B. $(4 x-1)(16 x^2-4 x+1)$
C. $(4 x-1)(16 x^2+4 x+1)$
D. $(4 x+1)(16 x^2+4 x+1)$
Answer (1)
Recognize the expression as a difference of cubes: ( 4 x ) 3 − 1 3 .
Apply the difference of cubes factorization formula: a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) with a = 4 x and b = 1 .
Substitute and simplify: ( 4 x − 1 ) (( 4 x ) 2 + ( 4 x ) ( 1 ) + 1 2 ) = ( 4 x − 1 ) ( 16 x 2 + 4 x + 1 ) .
The factored expression is ( 4 x − 1 ) ( 16 x 2 + 4 x + 1 ) .
Explanation
Recognizing the Difference of Cubes We are asked to factor the expression 64 x 3 − 1 . This expression is a difference of cubes, which can be factored using the formula a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) .
Identifying a and b In our case, we can rewrite 64 x 3 − 1 as ( 4 x ) 3 − 1 3 . Thus, a = 4 x and b = 1 .
Applying the Formula Now, we substitute a = 4 x and b = 1 into the difference of cubes formula: ( 4 x ) 3 − 1 3 = ( 4 x − 1 ) (( 4 x ) 2 + ( 4 x ) ( 1 ) + 1 2 ) Simplifying the expression, we get: ( 4 x − 1 ) ( 16 x 2 + 4 x + 1 )
Final Answer Comparing our result with the given options, we find that the correct factorization is ( 4 x − 1 ) ( 16 x 2 + 4 x + 1 ) .
Examples
Factoring the difference of cubes is useful in simplifying algebraic expressions and solving equations. For example, consider a scenario where you need to find the roots of the equation 64 x 3 − 1 = 0 . By factoring the left side as ( 4 x − 1 ) ( 16 x 2 + 4 x + 1 ) = 0 , you can easily find one real root, x = 4 1 . The quadratic factor 16 x 2 + 4 x + 1 has no real roots, as its discriminant is negative. This technique is also used in calculus for simplifying integrals and derivatives.
