Simplifying Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operators (such as +, -, ×, and /).

For example: 3x + 5 − 2x

• Terms: Parts separated by operators= 3x, 5, −2x
• Like terms: Terms with the same variable = 3x and −2x
• Constant: A term without a variable = 5

Combining Like Terms

Like terms are terms that have the same variable raised to the same power. To simplify an expression, combine these terms by performing the arithmetic operations.

Example: Simplify the expression 4x + 7 − 3x + 2.

• Combine 4x and −3x: 4x − 3x = x
• Combine 7 and 2: 7 + 2 = 9
• The simplified expression is x + 9.

Applying the Distributive Property

The distributive property states that a(b + c) = ab + ac. Use this property to eliminate parentheses in an expression.

Example: Simplify the expression 3(x + 4).

• Apply the distributive property: 3(x + 4) = 3x + 3×4
• Perform the multiplication: 3x + 12

The simplified expression is 3x+12.

Combining Distributive Property and Like Terms

In more complex expressions, use both the distributive property and combining like terms to simplify.

Example: Simplify the expression 2(x − 3) + 4x.

Apply the distributive property: 2(x − 3) = 2x − 6

Combine with 4x: 2x − 6 + 4x

Combine like terms: 2x + 4x = 6x

Simplified expression: 6x − 6

Using the FOIL Method

The FOIL method is used to simplify the product of the two binomials. FOIL stands for First, Outer, Inner, and Last, referring to the terms to be multiplied.

Example: Simplify (x + 2)(x + 3) using the FOIL method.

• First: x ⋅ x = x²
• Outer: x⋅3 = 3x
• Inner: 2⋅x = 2x
• Last: 2⋅3 = 6

Combine the results: x² + 3x + 2x + 6

Combine like terms: x² + 5x + 6

The simplified expression is x² + 5x + 6.

Solved Examples

Example 1: Simplify 4x + 3 − 2x + 5.

Combine like terms: 4x − 2x + 3 + 5.

Perform the addition/subtraction: 2x + 8.

The simplified expression is 2x + 8.

Example 2: Simplify 3(a+4)−2(a−1).

Apply the distributive property: 3(a + 4) = 3a + 12

−2(a − 1) = −2a + 2

Combine the results: 3a + 12 − 2a + 2

Combine like terms: 3a − 2a + 12 + 2.

Perform the addition/subtraction: a + 14.

The simplified expression is a + 14.

Example 3: Simplify \frac{6x^2 - 3x + 2x^2 + 5}{2}.

Combine like terms in the numerator:

6x^2 + 2x^2 - 3x + 5 = 8x^2 - 3x + 5

Divide each term by 2:

\frac{8x^2}{2} - \frac{3x}{2} + \frac{5}{2} = 4x^2 - \frac{3x}{2} + \frac{5}{2}

The simplified expression is 4x^2 - \frac{3x}{2} + \frac{5}{2}.

Example 4: Simplify (x + 2)(x - 3).

Use the FOIL method:

• First: x \cdot x = x^2
• Outer: x \cdot (-3) = -3x
• Inner: 2 \cdot x = 2x
• Last: 2 \cdot (-3) = -6

Combine the results:

x^2 - 3x + 2x - 6 = x^2 - x - 6

The simplified expression is x^2 - x - 6.

Example 5: Simplify \frac{2(x + 4) - 3(x - 2)}{x}.

Apply the distributive property in the numerator:

2(x + 4) = 2x + 8

-3(x - 2) = -3x + 6

Combine the results:

2x + 8 - 3x + 6 = -x + 14

Divide by x: \frac{-x + 14}{x} = -1 + \frac{14}{x}

The simplified expression is -1 + \frac{14}{x}.

Practice Questions

Q1: Simplify: 7m - 4n + 2m - 3n.

Q2: Simplify: \frac{5x^2 - 2x + 3x^2 - 4}{3}.

Q3: Simplify: (2x + 1)(x - 2) + 3(x + 1).

Q4: Simplify: \frac{4y - 3(2y - 1)}{y}.

Q5: Simplify: 5(a + 3b) - 2(a - b).

Q6: Simplify: (3x - 4)^2 - (x - 2)^2.

Q7: Simplify: 6 - 2(3 - x) + 4x.

Q8: Simplify: \frac{3(x^2 - 4) + 2(x^2 + 1)}{x}.

Q9: Simplify: (x + 2)(x + 3) - (x - 1)(x - 2).

Q10: Simplify: 2(3x - 5) + 4x - 7x.