Question
4. [-/1 Points] DETAILS MY NOTES TGEIALG6 5.8.074. Perform the division. (Simplify your answer completely.) \( \frac{x^{3}-8}{x-2} \) P
Ask by Flynn Bob. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \frac{x^{3}-8}{x-2} = x^{2}+2x+4 \)
Solution
Calculate or simplify the expression \( (x^3-8)/(x-2) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(x^{3}-8\right)}{\left(x-2\right)}\)
- step1: Remove the parentheses:
\(\frac{x^{3}-8}{x-2}\)
- step2: Calculate:
\(\frac{\left(x-2\right)\left(x^{2}+2x+4\right)}{x-2}\)
- step3: Reduce the fraction:
\(x^{2}+2x+4\)
The result of the division \( \frac{x^{3}-8}{x-2} \) is \( x^{2}+2x+4 \).
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Simplify this solution Bonus Knowledge
To simplify the expression \( \frac{x^{3}-8}{x-2} \), you can start by factoring the numerator \( x^{3}-8 \) using the difference of cubes formula, which states \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, with \( a = x \) and \( b = 2 \), you can factor it as: \[ x^{3}-8 = (x-2)(x^{2}+2x+4) \] Now, you substitute this back into the original division: \[ \frac{(x-2)(x^{2}+2x+4)}{x-2} \] Notice that you can cancel out \( x-2 \) from the numerator and denominator (as long as \( x \neq 2 \)): \[ x^{2}+2x+4 \] Thus, the simplified result is: \[ x^{2}+2x+4 \] So there you have it, a beautifully simplified expression!
