Factor each polynomial completely. See Example 6. 8t³ + 125

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Recognize that the polynomial $8t^{3} + 125$ is a sum of cubes because $8t^{3} = (2t)^{3}$ and $125 = 5^{3}$.

Recall the sum of cubes factoring formula: $a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})$.

Identify $a = 2t$ and $b = 5$ in the expression $8t^{3} + 125$.

Apply the formula: write the factorization as $(2t + 5)((2t)^{2} - (2t)(5) + 5^{2})$.

Simplify the terms inside the second parenthesis to get $(2t + 5)(4t^{2} - 10t + 25)$.

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Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). It is used to factor expressions where two terms are both perfect cubes added together. Recognizing 8t³ and 125 as cubes (2t)³ and 5³ allows applying this formula to factor the polynomial.

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A perfect cube is a number or expression raised to the third power, such as 8 = 2³ or t³. Identifying each term as a perfect cube is essential before applying the sum or difference of cubes formulas. This step ensures the correct factorization method is used.

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