(p2+p−6)(p2−6)

Jasonbradley

2 min read

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Jun 16, 2024

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Factoring the Expression (p² + p - 6)(p² - 6)

This expression involves two quadratic factors. Let's break down how to factor it completely.

Factoring the First Quadratic

The first factor, (p² + p - 6), is a quadratic trinomial that can be factored using the following steps:

1. Find two numbers that multiply to -6 and add up to 1. These numbers are 3 and -2.
2. Rewrite the middle term (p) using these numbers: (p² + 3p - 2p - 6)
3. Factor by grouping: p(p + 3) - 2(p + 3)
4. Factor out the common factor (p + 3): (p + 3)(p - 2)

Therefore, (p² + p - 6) factors to (p + 3)(p - 2).

Factoring the Second Quadratic

The second factor, (p² - 6), is a difference of squares. We can factor it using the following pattern:

a² - b² = (a + b)(a - b)

In this case, a = p and b = √6. So, the factored form is:

(p + √6)(p - √6)

Final Factored Form

Combining the factored forms of both quadratic factors, we get the completely factored expression:

(p + 3)(p - 2)(p + √6)(p - √6)

This is the final factored form of (p² + p - 6)(p² - 6).