X 3 X 5 Expand And Simplify

How to Expand and Simplify (x+3)(x+5): A Complete Guide

Mastering the art of expanding and simplifying algebraic expressions is a foundational skill that unlocks doors to more advanced mathematics, from calculus to engineering. At its core, this process transforms a compact, factored form like (x+3)(x+5) into a standard polynomial, revealing its true structure and making it ready for analysis, solving, or application. This seemingly simple task is far more than a mechanical exercise; it is the key to understanding how algebraic expressions behave, how their graphs are shaped, and how they model real-world phenomena. Whether you're a student building confidence or someone revisiting math fundamentals, this guide will walk you through the expand and simplify process with clarity, depth, and practical insight, ensuring you not only get the correct answer but truly comprehend the "why" behind every step.

Understanding the Starting Point: The Factored Form

The expression (x+3)(x+5) is in a factored form. So this means it is written as a product of two binomials—expressions with two terms each. The parentheses indicate multiplication. This form is incredibly useful for quickly finding the roots or zeros of the equation (the values of x that make the whole expression equal to zero). Also, from the factored form, we can immediately see that if x = -3 or x = -5, then one of the factors becomes zero, and thus the entire product is zero. Still, to perform operations like addition, subtraction, or comparison with other polynomials, or to analyze the shape of its graph, we need the expanded form. Here's the thing — expanding is the process of removing the parentheses by multiplying every term in the first binomial by every term in the second binomial. Still, simplification is the subsequent step of combining any like terms—terms that have the exact same variable part (e. g., x² terms with x², x terms with x).

The Step-by-Step Expansion Process: The FOIL Method

For the specific case of multiplying two binomials, the most reliable and memorable technique is FOIL. Still, this acronym stands for First, Outer, Inner, Last, describing the order in which we multiply the terms. It’s a systematic application of the distributive property.

Let’s apply FOIL to (x+3)(x+5):

1. First: Multiply the first term of the first binomial (x) by the first term of the second binomial (x).

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   • x * x = x²

2. Outer: Multiply the outer terms of the expression (the first term of the first binomial and the second term of the second binomial).

   • x * 5 = 5x

3. Inner: Multiply the inner terms (the second term of the first binomial and the first term of the second binomial).

   • 3 * x = 3x

4. Last: Multiply the last terms of each binomial.

   • 3 * 5 = 15

After performing these four multiplications, we write them all out as a sum: x² + 5x + 3x + 15

This is the expanded form, but it is not yet fully simplified.

The Crucial Simplification Step: Combining Like Terms

Now we look at our expanded expression: x² + 5x + 3x + 15. In practice, we identify like terms. The terms 5x and 3x are like terms because they both contain the variable x raised to the first power (x¹).

The x² term and the constant 15 have no other terms to combine with. Which means, the fully simplified expression is: x² + 8x + 15

This final trinomial (a polynomial with three terms) is the standard, simplified form of the original product (x+3)(x+5).

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