Problem 65 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the first section of the next chapter. $$\text { Add: } 5 x^{3}+12 x^{3}$$

Step-by-Step Solution

Answer

The result of \(5x^{3}+12x^{3}\) is \(17x^{3}\).

Step 1: Identification

First, identify the like terms in the expression. In this case, the like terms are \(5x^{3}\) and \(12x^{3}\), because they both have the same variable part, that is \(x^{3}\).

Step 2: Addition

Add the coefficients of the like terms. The coefficient of \(5x^{3}\) is 5, and the coefficient of \(12x^{3}\) is 12. Their sum is \(5 + 12 = 17\).

Step 3: Substitution

Substitute the sum of the coefficients back with the common variable part. This gives \(17x^{3}\).

Key Concepts

Like Terms in Algebra

Understanding 'like terms' in algebra is fundamental when dealing with polynomial addition. Like terms are terms that contain the same variable raised to the same power. For instance, in the expression 3a^2 + 7a^2, both terms are considered like terms because they contain the same variable, a, raised to the same power, 2. The coefficients (3 and 7 in this case) can be different.

Appreciating the concept of like terms lies at the heart of simplifying algebraic expressions. It's crucial to recognize that only the coefficients of like terms can be combined; their exponents do not change during the addition process. When looking at the original exercise, 5x^3 + 12x^3, we can see that both terms are like terms because they share the identical variable and exponent, x^3. This understanding is the first step to performing polynomial addition correctly.

Combining Like Terms

Once like terms are identified, the next step is to combine them, which is essentially a process of addition or subtraction. Combining like terms is straightforward: you simply add or subtract the coefficients while keeping the variable part unchanged. For example, consider the expression 4m + 9m. Both terms have m as the variable with no exponent, which means they are like terms and can be combined to yield 13m.

In the exercise provided, we combine the terms 5x^3 and 12x^3 by adding their coefficients (5 and 12) to get 17, while the x^3 remains the same. Thus, the resulting term is 17x^3. It's essential to remember that we can only combine terms with exactly matching variable parts; otherwise, they stay separate. This tactic not only applies to simple cases like the given exercise but also to more complex expressions involving multiple terms and variables.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division). These expressions do not have an equal sign, differentiating them from algebraic equations. An expression can be as simple as 7b or as complex as 2x^2 + 4y - 5/z.

The primary goal when working with algebraic expressions is simplification, where one combines like terms and performs arithmetic operations to condense the expression into its most reduced form. In the context of the provided exercise, the expression is simple, involving the addition of two like terms. However, remember that algebraic expressions can get quite intricate. Simplification helps in understanding the structure of these expressions and in preparing you for problem-solving that might involve substitution, factoring, or expanding algebraic expressions.