Factor. [tex]x^3-64 c^3[/tex]
Answer (2)
Recognize the expression as a difference of cubes: x 3 − 64 c 3 = x 3 − ( 4 c ) 3 .
Apply the difference of cubes formula: a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) with a = x and b = 4 c .
Substitute and simplify: ( x − 4 c ) ( x 2 + x ( 4 c ) + ( 4 c ) 2 ) = ( x − 4 c ) ( x 2 + 4 c x + 16 c 2 ) .
The factored form is: ( x − 4 c ) ( x 2 + 4 c x + 16 c 2 ) .
Explanation
Recognizing the Difference of Cubes We are asked to factor the expression x 3 − 64 c 3 . This expression is a difference of cubes. We can use the formula for the difference of cubes, which is a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) .
Rewriting the Expression In our case, we have x 3 − 64 c 3 . We can rewrite 64 c 3 as ( 4 c ) 3 . So, our expression becomes x 3 − ( 4 c ) 3 . Now we can apply the difference of cubes formula with a = x and b = 4 c .
Applying the Formula Using the formula, we get ( x − 4 c ) ( x 2 + x ( 4 c ) + ( 4 c ) 2 ) .
Simplifying the Expression Now we simplify the expression: ( x − 4 c ) ( x 2 + 4 c x + 16 c 2 ) .
Final Factored Form The factored form of x 3 − 64 c 3 is ( x − 4 c ) ( x 2 + 4 c x + 16 c 2 ) .
Examples
Factoring polynomials like x 3 − 64 c 3 is useful in many areas of mathematics and engineering. For example, when designing structures, engineers use polynomial equations to model the behavior of materials under stress. Factoring these polynomials can help identify critical points where the structure might be more susceptible to failure. Similarly, in signal processing, factoring polynomials is used to analyze and design filters that remove unwanted noise from signals. Understanding how to factor polynomials allows engineers to create more reliable and efficient systems.
The expression x 3 − 64 c 3 can be factored as ( x − 4 c ) ( x 2 + 4 c x + 16 c 2 ) by recognizing it as a difference of cubes and applying the relevant formula. This involves rewriting 64 c 3 as ( 4 c ) 3 and then using the difference of cubes formula. The final result gives us a clear factored expression.
;
