(x+5)(x^2-3x+1)

Jasonbradley

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Jun 17, 2024

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Expanding the Expression (x+5)(x^2-3x+1)

This article focuses on expanding the given algebraic expression: (x+5)(x^2-3x+1). This process is crucial in simplifying expressions and is often used in solving equations and inequalities.

Understanding the Process

The expression represents the product of two factors:

• (x+5): A binomial with two terms.
• (x^2-3x+1): A trinomial with three terms.

To expand this expression, we employ the distributive property. This involves multiplying each term in the first factor by each term in the second factor.

Step-by-Step Expansion

1. Distribute the first term of the first factor (x):

   • x * (x^2-3x+1) = x^3 - 3x^2 + x

2. Distribute the second term of the first factor (5):

   • 5 * (x^2-3x+1) = 5x^2 - 15x + 5

3. Combine the results from steps 1 and 2:

   • x^3 - 3x^2 + x + 5x^2 - 15x + 5

4. Simplify by combining like terms:

   • x^3 + 2x^2 - 14x + 5

Final Result

Therefore, the expanded form of (x+5)(x^2-3x+1) is x^3 + 2x^2 - 14x + 5.

Applications

Expanding expressions like this is essential in various mathematical contexts:

• Solving equations: Often, equations involve products of expressions. Expanding them allows for simplifying and solving for the unknown variable.
• Graphing functions: Expanding expressions can help determine the shape and behavior of graphs representing polynomial functions.
• Calculus: Expanding expressions is crucial in differentiating and integrating functions.

This simple process of expanding algebraic expressions is a fundamental skill in mathematics, playing a crucial role in various areas of study.