(x+5)(x^2-5x+25)

Jasonbradley

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Jun 17, 2024

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Expanding (x + 5)(x² - 5x + 25)

This expression represents the multiplication of a binomial (x + 5) and a trinomial (x² - 5x + 25). We can expand it using the distributive property or by recognizing a specific pattern.

Using the Distributive Property

1. Multiply the first term of the binomial by each term of the trinomial: x * (x² - 5x + 25) = x³ - 5x² + 25x

2. Multiply the second term of the binomial by each term of the trinomial: 5 * (x² - 5x + 25) = 5x² - 25x + 125

3. Combine the results from step 1 and step 2: x³ - 5x² + 25x + 5x² - 25x + 125

4. Simplify by combining like terms: x³ + 125

Therefore, the expanded form of (x + 5)(x² - 5x + 25) is x³ + 125.

Recognizing a Pattern

The trinomial (x² - 5x + 25) is a special form known as the sum of cubes pattern. This pattern states that:

(a + b)(a² - ab + b²) = a³ + b³

In our case, a = x and b = 5. Applying the pattern, we directly get:

(x + 5)(x² - 5x + 25) = x³ + 5³ = x³ + 125

This approach provides a quicker solution, but it's important to recognize the pattern for it to be applicable.

Conclusion

Expanding (x + 5)(x² - 5x + 25) results in x³ + 125. This can be achieved using the distributive property or recognizing the sum of cubes pattern. Both methods lead to the same result.