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Expanding the Expression (x + 5)(x + 2)
The expression (x + 5)(x + 2) represents the product of two binomials. To expand this expression, we can use the FOIL method, which stands for First, Outer, Inner, Last.
Here's how it works:
1. First: Multiply the first terms of each binomial: x * x = x²
2. Outer: Multiply the outer terms of the binomials: x * 2 = 2x
3. Inner: Multiply the inner terms of the binomials: 5 * x = 5x
4. Last: Multiply the last terms of each binomial: 5 * 2 = 10
Now we have: x² + 2x + 5x + 10
Finally, combine the like terms:
x² + 7x + 10
Therefore, the expanded form of (x + 5)(x + 2) is x² + 7x + 10.
Why is the FOIL Method Useful?
The FOIL method provides a systematic way to multiply binomials, ensuring that all terms are accounted for. It helps us to avoid mistakes and simplifies the process of expansion.
Applications of Expanding Binomials
Expanding binomials is a fundamental concept in algebra with various applications, including:
- Solving quadratic equations: By factoring a quadratic equation into two binomials, we can easily find its roots.
- Graphing quadratic functions: Expanding the expression helps us determine the vertex, axis of symmetry, and other key features of the parabola.
- Simplifying expressions: Expanding binomials can help simplify complex expressions and make them easier to work with.
By understanding the process of expanding binomials, we gain a deeper understanding of algebraic operations and their applications in various mathematical contexts.