(x%2B2).jpeg "(x-7)(x+2)")
Factoring and Expanding: (x-7)(x+2)
In algebra, we often encounter expressions that need to be factored or expanded. One such example is the expression (x-7)(x+2). Let's explore how to handle this expression:
Expanding the Expression
Expanding the expression means multiplying out the terms within the parentheses. We can do this using the FOIL method (First, Outer, Inner, Last):
- First: x * x = x²
- Outer: x * 2 = 2x
- Inner: -7 * x = -7x
- Last: -7 * 2 = -14
Combining these terms, we get: x² + 2x - 7x - 14
Simplifying further, we obtain the expanded form: x² - 5x - 14
Factoring the Expression
Factoring is the reverse process of expanding. We aim to find two binomials that, when multiplied, give us the original expression.
1. Finding the Leading Term:
The leading term of the expression is x². To get x² in our factored form, we need x as the first term in both binomials: (x )(x )
2. Finding the Constant Term:
The constant term is -14. We need to find two numbers that multiply to -14. Some possible pairs are:
- 1 and -14
- 2 and -7
- -1 and 14
- -2 and 7
3. Finding the Middle Term:
We need the pair of numbers from step 2 that, when added, give us the coefficient of the middle term (-5). The pair -7 and 2 satisfy this condition: -7 + 2 = -5.
4. Completing the Factored Form:
Therefore, the factored form of the expression is: (x - 7)(x + 2)
Conclusion:
We have demonstrated how to both expand and factor the expression (x-7)(x+2). Understanding these processes is crucial for solving algebraic equations and simplifying expressions in various mathematical contexts.