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(5x-9)(x-2)(x+4)

5 min read Jun 16, 2024
(5x-9)(x-2)(x+4)

Factoring and Expanding the Expression: (5x-9)(x-2)(x+4)

This article will explore the process of factoring and expanding the expression (5x-9)(x-2)(x+4). We'll cover both the expansion of this expression into a polynomial and the steps involved in factoring it back into its original form.

Expanding the Expression

To expand this expression, we'll use the distributive property multiple times.

  1. Start with the first two factors: (5x-9)(x-2)

    Expanding this gives us:

    (5x-9)(x-2) = 5x(x-2) - 9(x-2)
            = 5x^2 - 10x - 9x + 18
            = 5x^2 - 19x + 18

  2. Now multiply the result by the remaining factor: (x+4)

    Expanding this gives us:

    (5x^2 - 19x + 18)(x+4) = 5x^2(x+4) - 19x(x+4) + 18(x+4)
                       = 5x^3 + 20x^2 - 19x^2 - 76x + 18x + 72
                       = **5x^3 + x^2 - 58x + 72**

Therefore, the expanded form of (5x-9)(x-2)(x+4) is 5x^3 + x^2 - 58x + 72.

Factoring the Expression

Factoring the expanded polynomial back into its original form is a bit more challenging. Here's a breakdown of the process:

  1. Identify potential factors: We need to find three factors that multiply to give us the original expression.

    • Leading coefficient: The leading coefficient of the polynomial (5) is a hint that one of the factors might have a leading coefficient of 5.
    • Constant term: The constant term (72) provides clues about the constant terms of the factors. We need to find combinations of numbers that multiply to 72.

  2. Trial and error: We can try different combinations of factors using the information from step 1.

    • Since the leading coefficient of the original expression was 5, we'll try a factor of (5x - a) where 'a' is a constant.
    • We also know that the constant terms of the factors need to multiply to 72.
    • We might notice that (x-2) is a factor, which was one of the original factors. This gives us a starting point for testing.

  3. Check for the remaining factor: With (5x-9) and (x-2) as factors, we can divide the expanded polynomial by their product. This will give us the remaining factor (x+4).

By using a combination of observation, trial and error, and understanding the relationship between the factors and the expanded polynomial, we can factor the expression back to its original form: (5x-9)(x-2)(x+4).

Conclusion

This article demonstrates the process of expanding and factoring the expression (5x-9)(x-2)(x+4). Expanding the expression involves using the distributive property, while factoring requires careful observation, trial and error, and understanding the relationships between the coefficients and the factors.

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