Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression, or explain why the expression cannot be simplified. $$10 x^{3}+5 x^{3}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( 15x^3 \).

Step 1: Identify like terms

Here, like terms are \( 10x^3 \) and \( 5x^3 \). Both are multiples of \( x^3 \).

Step 2: Combine like terms

Add the coefficients of the like terms, 10 and 5, together. So the calculation is \( 10x^3 + 5x^3 \) which equals \( 15x^3 \).

Key Concepts

Like Terms

When working with algebraic expressions, the term "like terms" refers to those parts of the expression that have the exact same variables raised to the same powers. In the expression \(10x^3 + 5x^3\), both \(10x^3\) and \(5x^3\) are like terms because they both include the variable \(x\) raised to the third power.

For terms to be considered "like," they should:

• Have the same variable(s)
• Have those variables raised to the same exponent(s)

This consistency allows them to be combined or simplified as a single term. Identifying like terms is crucial because it helps simplify expressions, enabling us to perform calculations more easily. Remember, even if the coefficients—the numbers in front of the variables—are different, as long as the variables and their exponents match, the terms are considered like terms.

Coefficients

In algebra, a coefficient is the numerical part of a term that contains a variable. For example, in the term \(10x^3\), the coefficient is 10. Similarly, in \(5x^3\), the coefficient is 5. Coefficients represent how many times the variable part is being multiplied.

Understanding coefficients is important because:

• They are key in combining like terms, allowing us to add or subtract terms easily.
• They can impact the overall value of a term when the variable is substituted with a numerical value.

In our expression \(10x^3 + 5x^3\), you focus on the coefficients 10 and 5 to combine these terms. We retain the variable part \(x^3\), and simply add their coefficients to simplify the expression.

Combining Like Terms

Now that you've identified like terms and understood coefficients, the next step is combining like terms. This process involves adding or subtracting the coefficients of terms that have like variables and exponents. By combining like terms, we make the expression simpler and usually easier to work with.

In the algebraic expression \(10x^3 + 5x^3\):

• Both terms are like terms because they consist of \(x^3\).
• Add their coefficients together: \(10 + 5 = 15\).
• Retain the common variable and exponent: the result is \(15x^3\).

Combining like terms is a powerful tool because it condenses expressions and prepares them for further algebraic operations, such as factoring or solving equations. Simplification through this method helps maintain an organized and clear approach in evaluating and solving algebraic expressions.