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Expanding the Expression (x+4)(x+5)
This expression represents the product of two binomials, (x+4) and (x+5). To expand it, we can use the FOIL method. FOIL stands for First, Outer, Inner, Last, which helps us remember the order of multiplying terms:
1. First: Multiply the first terms of each binomial: x * x = x² 2. Outer: Multiply the outer terms of the binomials: x * 5 = 5x 3. Inner: Multiply the inner terms of the binomials: 4 * x = 4x 4. Last: Multiply the last terms of each binomial: 4 * 5 = 20
Now we have: x² + 5x + 4x + 20
Finally, combine the like terms: x² + 9x + 20
Therefore, the expanded form of (x+4)(x+5) is x² + 9x + 20.
Understanding the FOIL Method:
The FOIL method provides a systematic way to expand binomials. It ensures that we multiply every term in the first binomial by every term in the second binomial, resulting in the correct expanded form.
Applications:
Expanding expressions like (x+4)(x+5) is fundamental in algebra and has various applications in:
- Solving equations: By expanding the expression, we can manipulate equations and solve for unknown variables.
- Graphing functions: The expanded form helps us understand the shape and behavior of quadratic functions represented by the expression.
- Factoring polynomials: Understanding how to expand expressions helps us to recognize and factor polynomials into simpler forms.
By mastering the FOIL method and understanding its applications, you can confidently work with algebraic expressions and explore more complex mathematical concepts.