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Find the binomial that completes the factorization. $\newline$ $t^3 - 125 = (\underline{\hspace{3em}})(t^2 + 5t + 25)$

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Factor sums and differences of cubes

Full solution

Q. Find the binomial that completes the factorization.

\newline

t^3 - 125 = (\underline{\hspace{3em}})(t^2 + 5t + 25)

Recognize Cubes Difference: Recognize that $t^3 - 125$ is a difference of cubes since $125$ is $5^3$ .
Apply Formula: Use the formula for factoring a difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .
Identify Values: Identify $a = t$ and $b = 5$ to apply the formula.
Plug into Formula: Plug $a$ and $b$ into the formula: $(t - 5)(t^2 + 5t + 25)$ .
Check Trinomial Match: Check if the second factor matches the given trinomial $t^2 + 5t + 25$ .

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