Problem 30 - Algebra A Combined Function

Question

Simplify each expression by combining like terms. See Examples 6 through 10. $$ 18 x^{3}+4 x^{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(22x^3\).

Step 1: Identify Like Terms

In the expression \(18x^3 + 4x^3\), notice that both terms have the variable \(x\) raised to the power of 3. These are 'like terms' because they have the same variable and exponent.

Step 2: Combine Coefficients

To simplify, add the coefficients of the like terms together. Add the coefficient \(18\) from the first term to the coefficient \(4\) from the second term. This gives you \(18 + 4 = 22\).

Step 3: Write the Simplified Expression

The simplified expression becomes \(22x^3\) because you add the coefficients together while keeping the variable \(x^3\) the same.

Key Concepts

Polynomials

Polynomials are expressions made up of terms that involve variables, exponents, and coefficients. Each term in a polynomial can include:

• A variable, which is often represented by letters like \( x \), \( y \), or \( z \).
• An exponent, which shows the power to which the variable is raised.
• A coefficient, which is the numerical factor multiplying the variable term.

Polynomials can have different numbers of terms: a single term (monomial), two terms (binomial), or more (polynomial). Understanding the structure of polynomials is essential, especially when you need to simplify or factor expressions. In our exercise, we dealt with a polynomial made of two terms: \( 18x^3 \) and \( 4x^3 \). Both terms involve the same variable \( x \) raised to the same power, which allows us to combine them easily.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often includes:

• Combining like terms, which are terms with the same variable and exponent.
• Performing arithmetic operations on coefficients while keeping the variable parts unchanged.

In the example \( 18x^3 + 4x^3 \), both terms are like terms as they have \( x^3 \). Therefore, they can be simplified by adding their coefficients. Simplifying helps to make expressions easier to understand and solve. It's especially useful when you want to solve equations, graph a polynomial, or simply make the expression neat. Once you have combined the coefficients, you should always double-check the expression to ensure it is simplified correctly.

Coefficients

Coefficients are crucial elements in algebraic expressions. They dictate the size or scale of a term and provide information on how much of a variable is present. Here's what you need to know:

• The coefficient is the numerical part of a term, standing in front of the variable.
• In the term \( 18x^3 \), \( 18 \) is the coefficient. In \( 4x^3 \), \( 4 \) is the coefficient.
• When combining like terms, you add the coefficients together.

In our specific exercise, adding the coefficients \( 18 \) and \( 4 \) results in \( 22 \), forming the simplified term \( 22x^3 \). This step is critical in the process of simplifying expressions and understanding the role of coefficients ensures that you can manipulate expressions correctly.