(x%2Bb)-examples.jpeg "(x+a)(x+b) Examples")
Expanding (x + a)(x + b)
The expression (x + a)(x + b) is a common algebraic form that often appears in various mathematical contexts. It represents the product of two binomials, where 'x' is a variable and 'a' and 'b' are constants. Expanding this expression involves multiplying each term in the first binomial with each term in the second binomial.
Understanding the Process
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FOIL Method: One popular way to expand this expression is using the FOIL method, which stands for First, Outer, Inner, Last.
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms: x * b = bx
- Inner: Multiply the inner terms: a * x = ax
- Last: Multiply the last terms: a * b = ab
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Distributive Property: Alternatively, we can use the distributive property to expand the expression. This involves distributing each term in the first binomial over the second binomial:
- x(x + b) + a(x + b)
Then, we apply the distributive property again:
- x² + bx + ax + ab
Combining Like Terms
After applying either method, we obtain the expanded expression: x² + bx + ax + ab.
Note: The terms bx and ax are like terms because they both contain the variable 'x'. We can combine these terms to simplify the expression further:
- Simplified expression: x² + (a + b)x + ab
Examples
Let's illustrate this with some examples:
Example 1: (x + 2)(x + 3)
- FOIL Method: x² + 3x + 2x + 6
- Simplified expression: x² + 5x + 6
Example 2: (x - 5)(x + 4)
- FOIL Method: x² + 4x - 5x - 20
- Simplified expression: x² - x - 20
Example 3: (2x + 1)(3x - 2)
- FOIL Method: 6x² - 4x + 3x - 2
- Simplified expression: 6x² - x - 2
Conclusion
Expanding the expression (x + a)(x + b) is a fundamental skill in algebra. Mastering the FOIL method and the distributive property allows us to efficiently simplify and manipulate algebraic expressions. The simplified expression x² + (a + b)x + ab provides a concise and general form for this type of product.