(x^3+x^2-9x-6)/(x^2-9)

Jasonbradley

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Jun 17, 2024

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Factoring and Simplifying the Expression (x^3 + x^2 - 9x - 6) / (x^2 - 9)

This article explores the process of simplifying the rational expression (x^3 + x^2 - 9x - 6) / (x^2 - 9). We will achieve this by factoring both the numerator and denominator and then canceling out any common factors.

Factoring the Numerator

The numerator, x^3 + x^2 - 9x - 6, can be factored by grouping:

1. Group the terms: (x^3 + x^2) + (-9x - 6)
2. Factor out the greatest common factor (GCF) from each group: x^2(x + 1) - 6(x + 1)
3. Factor out the common binomial factor: (x + 1)(x^2 - 6)

Therefore, the factored numerator is (x + 1)(x^2 - 6).

Factoring the Denominator

The denominator, x^2 - 9, is a difference of squares:

1. Recognize the pattern: x^2 - 9 = x^2 - 3^2
2. Apply the difference of squares formula: a^2 - b^2 = (a + b)(a - b)

Applying this formula, we get: (x + 3)(x - 3).

Simplifying the Expression

Now that both the numerator and denominator are factored, we can rewrite the expression:

(x^3 + x^2 - 9x - 6) / (x^2 - 9) = (x + 1)(x^2 - 6) / (x + 3)(x - 3)

We can see that there are no common factors to cancel out. This means the expression is already simplified in its factored form.

Conclusion

The simplified form of the expression (x^3 + x^2 - 9x - 6) / (x^2 - 9) is (x + 1)(x^2 - 6) / (x + 3)(x - 3). Remember that this expression is undefined for x = 3 and x = -3, as these values make the denominator zero.