how to factor x^3 - 125
Understand the Problem
The question is asking how to factor the expression x^3 - 125, which is a difference of cubes. We can solve this by applying the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, we identify a as x and b as 5.
Answer
$(x - 5)(x^2 + 5x + 25)$
Answer for screen readers
The factored form of the expression is $(x - 5)(x^2 + 5x + 25)$.
Steps to Solve
- Identify the values of a and b
Given the expression $x^3 - 125$, we set $a = x$ and $b = 5$ because $125$ is equal to $5^3$.
- Apply the difference of cubes formula
Using the difference of cubes formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, substitute $a$ and $b$:
$$ x^3 - 125 = (x - 5)(x^2 + x(5) + 5^2) $$
- Simplify the expression
Now we need to evaluate the quadratic term:
- Calculate $ab$: $x(5) = 5x$
- Calculate $b^2$: $5^2 = 25$
So the expression becomes:
$$ x^3 - 125 = (x - 5)(x^2 + 5x + 25) $$
The factored form of the expression is $(x - 5)(x^2 + 5x + 25)$.
More Information
The expression $x^3 - 125$ is known as a difference of cubes, which is a special factorization that can simplify polynomial expressions. Understanding this formula helps in solving higher-degree polynomials.
Tips
- A common mistake is misidentifying the values of $a$ and $b$. Ensure to recognize that $125$ is $5^3$.
- Another mistake is forgetting to simplify the $a^2 + ab + b^2$ term correctly. Make sure to calculate each component carefully.
