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(x-2)(x+3)(x-7) In Standard Form

2 min read Jun 17, 2024
(x-2)(x+3)(x-7) In Standard Form

Expanding and Simplifying (x-2)(x+3)(x-7)

This article will guide you through the process of expanding and simplifying the expression (x-2)(x+3)(x-7) to its standard form.

Step 1: Multiply the first two factors

Begin by multiplying the first two factors: (x-2)(x+3). This can be done using the FOIL method (First, Outer, Inner, Last) or by distributing.

  • FOIL:

    • F: x * x = x²
    • O: x * 3 = 3x
    • I: -2 * x = -2x
    • L: -2 * 3 = -6
    • Combining the terms, we get: x² + 3x - 2x - 6 = x² + x - 6

  • Distribution:

    • x(x+3) - 2(x+3) = x² + 3x - 2x - 6 = x² + x - 6

Therefore, (x-2)(x+3) = x² + x - 6

Step 2: Multiply the result by the remaining factor

Now we have: (x² + x - 6)(x-7)

Again, we can use the distributive property or the FOIL method:

  • Distribution:

    • x²(x-7) + x(x-7) - 6(x-7) = x³ - 7x² + x² - 7x - 6x + 42

  • FOIL:

    • F: x² * x = x³
    • O: x² * -7 = -7x²
    • I: x * x = x²
    • L: x * -7 = -7x
    • F: -6 * x = -6x
    • O: -6 * -7 = 42
    • Combining the terms, we get: x³ - 7x² + x² - 7x - 6x + 42

Step 3: Combine like terms

Finally, combine the like terms in the resulting expression:

x³ - 7x² + x² - 7x - 6x + 42 = x³ - 6x² - 13x + 42

Conclusion

The standard form of the expression (x-2)(x+3)(x-7) is x³ - 6x² - 13x + 42.

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