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Expanding and Simplifying (x+1)(x+3)
The expression (x+1)(x+3) represents the product of two binomials. To simplify this expression, we can use the FOIL method, which stands for First, Outer, Inner, Last.
Here's how it works:
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First: Multiply the first terms of each binomial: x * x = x²
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Outer: Multiply the outer terms of the binomials: x * 3 = 3x
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Inner: Multiply the inner terms of the binomials: 1 * x = x
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Last: Multiply the last terms of each binomial: 1 * 3 = 3
Now, combine all the terms: x² + 3x + x + 3
Finally, simplify by combining like terms: x² + 4x + 3
Therefore, the expanded and simplified form of (x+1)(x+3) is x² + 4x + 3.
Understanding the FOIL Method
The FOIL method is a helpful mnemonic for remembering the steps involved in multiplying binomials. However, it's essential to understand that it's just a visual representation of the distributive property.
The distributive property states that:
- a(b + c) = ab + ac
In our example, we can apply the distributive property twice:
- (x + 1)(x + 3) = (x + 1) * x + (x + 1) * 3
- = x² + x + 3x + 3
- = x² + 4x + 3
As you can see, both methods lead to the same result.
Applications of Expanding Binomials
Expanding binomials like (x+1)(x+3) has various applications in algebra and beyond:
- Factoring quadratics: Understanding the expanded form of binomials helps us factor quadratic expressions.
- Solving equations: Expanding binomials is crucial for solving equations involving quadratic expressions.
- Calculus: Binomial expansions play a significant role in calculus, especially when dealing with derivatives and integrals.
- Real-world problems: Many real-world situations can be modeled using quadratic equations, requiring us to expand binomials to solve them.
Understanding the basics of expanding and simplifying binomials is a crucial step in mastering algebra and other related fields.