Simplifying Like Terms

Introduction

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that contain the same variables raised to the same powers. The coefficients of these terms can differ, but the variable parts must be identical for them to be considered like terms. Simplifying expressions by combining like terms simplifies calculations and solves equations more efficiently.

For example, in the expression $3x + 5x$, both terms contain the variable $x$ raised to the first power. Therefore, they are like terms and can be combined to get $8x$. Conversely, in $3x + 5y$, the terms are not like terms because they contain different variables.

The Importance of Coefficients

The coefficient of a term is the numerical factor multiplying the variable(s). When simplifying like terms, it is crucial to correctly identify and manage these coefficients. Accurate manipulation of coefficients ensures that expressions are simplified correctly.

Consider the expression $4a + 5a - 2a$. Since all terms are like terms (all contain the variable $a$), you combine the coefficients: $4 + 5 - 2 = 7$. Therefore, the simplified form is $7a$.

The Role of Exponents

Exponents indicate the power to which a base is raised and play a significant role in determining whether terms are like terms. For terms to be like terms, not only must the variables be identical, but their exponents must also match.

For instance, in the expression $3x^2 + 4x$, the terms are not like terms because $x^2$ and $x$ have different exponents. Therefore, they cannot be combined through addition or subtraction.

Simplification Rules

To simplify like terms, follow these steps:

1. Identify like terms by ensuring both the variables and their exponents match.
2. Add or subtract the coefficients of these like terms.
3. Rewrite the expression with the combined coefficients and the common variable part.

For example, simplify the expression $2x + 3x - x$:

1. All terms are like terms since they all contain the variable $x$ with the same exponent.
2. Add the coefficients: $2 + 3 - 1 = 4$.
3. Rewrite the expression as $4x$.

Distributive Property and Like Terms

The distributive property allows us to expand expressions by distributing a factor across terms within parentheses. This property is instrumental in simplifying expressions and combining like terms.

Consider the expression $2(3x + 4)$. Applying the distributive property: $$ 2(3x + 4) = 2 \cdot 3x + 2 \cdot 4 = 6x + 8 $$ Here, $6x$ and $8$ are separate terms; however, they cannot be combined further as they are not like terms.

Combining Like Terms in Polynomials

Polynomials consist of multiple terms with varying degrees of variables. Simplifying polynomials involves combining like terms to reduce the expression to its simplest form.

For example, simplify the polynomial $4x^3 + 2x^2 - x + 5 + 3x^3 - x^2 + 2x - 3$:

1. Group like terms:
   • Terms with $x^3$: $4x^3 + 3x^3$
   • Terms with $x^2$: $2x^2 - x^2$
   • Terms with $x$: $-x + 2x$
   • Constant terms: $5 - 3$
2. Combine coefficients:
   • $4x^3 + 3x^3 = 7x^3$
   • $2x^2 - x^2 = x^2$
   • $-x + 2x = x$
   • $5 - 3 = 2$
3. Write the simplified polynomial: $7x^3 + x^2 + x + 2$

Applications in Solving Equations

Simplifying like terms is essential when solving algebraic equations. By reducing equations to their simplest form, we can isolate variables and find their values more efficiently.

For example, solve the equation $3x + 5x - 2 = 16$:

1. Combine like terms: $3x + 5x = 8x$.
2. The equation becomes $8x - 2 = 16$.
3. Add 2 to both sides: $8x = 18$.
4. Divide both sides by 8: $x = \frac{18}{8} = \frac{9}{4}$.

Common Mistakes to Avoid

When simplifying like terms, students often make mistakes related to:

• Mismatching variables or exponents: Combining terms that do not have identical variable parts.
• Incorrect arithmetic operations: Errors in adding or subtracting coefficients.
• Forgetting to apply the distributive property: Leading to incomplete simplification.

Being meticulous in identifying like terms and performing arithmetic operations accurately is crucial to avoid these common errors.

Advanced Applications

Simplifying like terms extends beyond basic algebra and is foundational in more advanced mathematical areas such as calculus, linear algebra, and polynomial factorization. Mastery of this skill enables students to tackle complex mathematical problems with confidence.

For instance, in calculus, simplifying expressions is necessary when finding derivatives or integrals of polynomial functions. In linear algebra, combining like terms is essential when performing matrix operations.

Example Problems

Let's explore a few examples to solidify the understanding of simplifying like terms:

1. Example 1: Simplify $7y - 3y + 2y$.
   • Combine coefficients: $7 - 3 + 2 = 6$.
   • Simplified expression: $6y$.
2. Example 2: Simplify $5a^2 + 3ab - 2a^2 + ab$.
   • Identify like terms:
     • $5a^2 - 2a^2 = 3a^2$
     • $3ab + ab = 4ab$
   • Simplified expression: $3a^2 + 4ab$.
3. Example 3: Simplify $2(x + 3) + 4(x - 2)$ using the distributive property.
   • Apply distributive property: $2x + 6 + 4x - 8$.
   • Combine like terms:
     • $2x + 4x = 6x$
     • $6 - 8 = -2$
   • Simplified expression: $6x - 2$.

Practice Exercises

Enhance your understanding by practicing the following exercises:

1. Simplify $9m - 4m + 2m$.
2. Simplify $3x^2 + 5x - 2x^2 + 7$.
3. Simplify $4(a + b) - 2(a - b)$.
4. Simplify $6p^3 + 2p^2 - p + 3 + p^3 - 4p^2 + 2p - 5$.

Answers:

1. $9m - 4m + 2m = 7m$
2. $3x^2 - 2x^2 + 5x + 7 = x^2 + 5x + 7$
3. $4a + 4b - 2a + 2b = 2a + 6b$
4. $6p^3 + p^3 + 2p^2 - 4p^2 - p + 2p + 3 - 5 = 7p^3 - 2p^2 + p - 2$

Comparison Table

Summary and Key Takeaways