(x-3).jpeg "(x-8)(x-3)")
Factoring and Expanding (x-8)(x-3)
This article will explore the process of factoring and expanding the expression (x-8)(x-3). We'll cover the basics of these operations and then apply them to this specific example.
Understanding the Basics
- Factoring is the process of breaking down a mathematical expression into simpler expressions that multiply together to give the original expression.
- Expanding is the opposite of factoring; it involves multiplying out expressions to simplify them into a single expression.
Expanding (x-8)(x-3)
To expand this expression, we'll use the FOIL method:
First: Multiply the first terms of each binomial: x * x = x² Outer: Multiply the outer terms: x * -3 = -3x Inner: Multiply the inner terms: -8 * x = -8x Last: Multiply the last terms: -8 * -3 = 24
Now, combine the terms: x² - 3x - 8x + 24
Finally, simplify by combining like terms: x² - 11x + 24
Therefore, the expanded form of (x-8)(x-3) is x² - 11x + 24.
Factoring x² - 11x + 24
We can now factor the expanded expression back into its original form. Here's how:
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Find two numbers that multiply to give the constant term (24) and add up to the coefficient of the middle term (-11). These numbers are -8 and -3.
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Rewrite the middle term (-11x) using these two numbers: x² - 8x - 3x + 24
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Factor by grouping:
- Group the first two terms and the last two terms: (x² - 8x) + (-3x + 24)
- Factor out the greatest common factor (GCF) from each group: x(x - 8) - 3(x - 8)
- Notice that both groups have a common factor of (x - 8): (x - 8)(x - 3)
This brings us back to our original expression!
Conclusion
We have successfully expanded and factored the expression (x-8)(x-3). This process demonstrates the relationship between factoring and expanding and highlights the FOIL method as a useful tool for simplifying expressions.