Combining Like Terms

Introduction

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have identical variables raised to the same power. They differ only in their coefficients. For example, in the expression $3x^2 + 5x - 2x^2 + 7$, the terms $3x^2$ and $-2x^2$ are like terms, while $5x$ and $7$ are not like terms with $x^2$ or each other.

The Importance of Combining Like Terms

Combining like terms simplifies algebraic expressions, making them easier to work with. This simplification is crucial when solving equations, factoring, and performing other algebraic operations. It reduces complexity and helps in identifying patterns and relationships within mathematical problems.

Steps to Combine Like Terms

1. Identify Like Terms: Look for terms with the same variable part. The variable parts must be identical in both the variable and its exponent.
2. Combine Coefficients: Add or subtract the coefficients of like terms while keeping the variable part unchanged.
3. Simplify the Expression: After combining, rewrite the expression in its simplified form.

Examples of Combining Like Terms

Consider the expression $4y + 3y - 2 + 5$. Here, $4y$ and $3y$ are like terms, and $-2$ and $5$ are constant terms that can also be combined.

Combining the like terms:

So, the simplified expression is:

Combining Like Terms with Variables Raised to Powers

When variables are raised to powers, only terms with the same variable and exponent are considered like terms. For example, $2x^3 + 5x^2 - x^3 + 4x^2$ can be simplified by combining like terms as follows:

The simplified expression is:

Combining Like Terms with Multiple Variables

In expressions with multiple variables, like terms must have the same variables raised to the same powers in the same order. For example, $3ab + 2a b - ab + 4ac$ combines the terms with $ab$.

Combining the like terms:

The term $4ac$ remains unchanged as it does not have the same variable components as $ab$.

The simplified expression is:

Using the Distributive Property to Combine Like Terms

The distributive property can be used to simplify expressions before combining like terms. For instance, consider the expression $2(x + 3) + 4x$.

First, apply the distributive property:

Now, combine like terms:

The simplified expression is:

Combining Like Terms in Polynomial Expressions

Polynomials often require combining like terms to simplify. For example, in the polynomial $5x^4 - 3x^2 + 2x^4 + x^2 - 7$, combine the $x^4$ and $x^2$ terms separately.

Combining like terms:

The simplified polynomial is:

Common Mistakes When Combining Like Terms

• Mismatching Variables or Exponents: Combining terms with different variables or exponents is incorrect. For example, $3x$ and $3y$ are not like terms.
• Ignoring Negative Signs: Failing to consider negative coefficients can lead to incorrect results. Carefully handle subtraction when combining terms.
• Overlooking Constants: Constants (terms without variables) should be combined separately from variable terms.

Practice Problems

1. Simplify the expression: $7m + 3n - 2m + 5n$.
2. Combine like terms: $4x^2 + 5x - 3x^2 + 2x - 8$.
3. Simplify: $6ab - 2ac + 4ab + 3ac$.
4. Combine the terms: $9y^3 + 2y^2 - y^3 + 4y^2$.

Solutions to Practice Problems

1. Simplify $7m + 3n - 2m + 5n$:

   $7m - 2m = 5m$
   $3n + 5n = 8n$
   

   Result:

   $$5m + 8n$$

2. Combine like terms in $4x^2 + 5x - 3x^2 + 2x - 8$:

   $4x^2 - 3x^2 = x^2$
   $5x + 2x = 7x$
   

   Result:

   $$x^2 + 7x - 8$$

3. Simplify $6ab - 2ac + 4ab + 3ac$:

   $6ab + 4ab = 10ab$
   $-2ac + 3ac = ac$
   

   Result:

   $$10ab + ac$$

4. Combine the terms $9y^3 + 2y^2 - y^3 + 4y^2$:

   $9y^3 - y^3 = 8y^3$
   $2y^2 + 4y^2 = 6y^2$
   

   Result:

   $$8y^3 + 6y^2$$

Comparison Table

Summary and Key Takeaways