How do you factor (p)^3 - 8c^3 when p = (a + b) and b = 2c?

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SUMMARY

The expression (p)^3 - 8c^3 can be factored using the difference of cubes formula, where p = (a + b) and b = 2c. The factorization results in (a + b - 2c)[(a + b)^2 + 2c(a + b) + 4c^2]. This method simplifies the expression effectively by substituting the values of a and b into the formula, ensuring clarity in the steps taken to reach the final factorization.

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mathdad
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Factor (a + b)^3 - 8c^3.

Difference of cubes, right?

I say in the formula, a = (a + b) and b = 2c.

Right?
 
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$$(a-b)^3=a^3-3a^2b+3ab^2-b^3$$

Rearrange:

$$\begin{align*}a^3-b^3&=(a-b)^3+3a^2b-3ab^2 \\
&=(a-b)^3+3ab(a-b) \\
&=(a-b)[(a-b)^2+3ab] \\
&=(a-b)(a^2-2ab+b^2+3ab) \\
&=(a-b)(a^2+ab+b^2)\end{align*}$$

$$(a+b)^3-8c^3=(a+b-2c)[(a+b)^2+2c(a+b)+4c^2]$$
 
RTCNTC said:
Factor (a + b)^3 - 8c^3.

Difference of cubes, right?

I say in the formula, a = (a + b) and b = 2c.

Right?
I mentioned this in another thread. Don't set the LHS of "a = a + b" It's too confusing.

-Dan
 
topsquark said:
I mentioned this in another thread. Don't set the LHS of "a = a + b" It's too confusing.

-Dan

I got it. I will let p = (a + b). I will then substitute into the difference of cubes formula and simplify as much as possible.
 

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