Simplifying Algebraic Expressions by Combining Like Terms
Simplifying an algebraic expression means rewriting it in a simpler form without changing its value. One common method is to combine like terms. Like terms are terms that have exactly the same variable parts raised to the same powers. For example, 2a and 5a are like terms because both contain the variable a to the first power, but 3b and 5a are not like terms because they have different variables.
To simplify, add or subtract the coefficients (numerical parts) of like terms while keeping the variable part unchanged.
Example: Simplify 2a + 3b + 5a - 9b. Identify like terms: 2a and 5a; 3b and -9b. Combine: 2a + 5a = 7a, 3b - 9b = -6b. So, the simplified expression is 7a - 6b.
- Identify like terms by matching variables and powers.
- Add or subtract coefficients of like terms.
- Rewrite the expression with combined terms only.
Expanding Products of Algebraic Expressions
Expansion means multiplying out brackets to remove them and write the expression as a sum or difference of terms. This is important for simplifying expressions and solving equations.
When expanding a single term multiplied by a bracket, multiply the term by each term inside the bracket.
Example: Expand 3x(2x - 4y). Multiply 3x by 2x to get 6x², and 3x by -4y to get -12xy. The expanded expression is 6x² - 12xy.
- Multiply the term outside the bracket by each term inside the bracket.
- Keep track of signs (positive or negative).
- Write all resulting terms without brackets.
Expanding the Product of Two Binomials
When multiplying two brackets, each term in the first bracket must be multiplied by each term in the second bracket. This is often remembered as FOIL (First, Outer, Inner, Last) for binomials.
Example: Expand (2x + 1)(x - 4). Multiply 2x by x (First) to get 2x², 2x by -4 (Outer) to get -8x, 1 by x (Inner) to get x, and 1 by -4 (Last) to get -4. Combine like terms: -8x + x = -7x. The expanded expression is 2x² - 7x - 4.
- Multiply each term in the first bracket by each term in the second bracket.
- Combine like terms after multiplication.
- Write the final expression without brackets.
Factoring Expressions by Extracting Common Factors
Factoring is the reverse of expansion. It involves writing an expression as a product of factors. One common method is to identify and extract the greatest common factor (GCF) from all terms.
To factor completely, find the largest algebraic expression (including numbers and variables) that divides all terms.
Example: Factor 9x + 15xy. The coefficients 9 and 15 have a GCF of 3. Both terms have at least one x. So, the GCF is 3x. Divide each term by 3x: 9x ÷ 3x = 3, 15xy ÷ 3x = 5y. Write the factored form as 3x(3 + 5y).
- Find the greatest common factor of coefficients and variables.
- Divide each term by the GCF to find the remaining factor.
- Write the expression as the product of the GCF and the remaining factor.
- Check by expanding to confirm correctness.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Simplifying | Combine like terms by adding/subtracting coefficients | 2a + 3b + 5a - 9b → 7a - 6b | Always group terms with the same variables before combining |
| Expanding single bracket | Multiply term outside bracket by each term inside | 3x(2x - 4y) → 6x² - 12xy | Show each multiplication step clearly to gain method marks |
| Expanding two brackets | Multiply each term in first bracket by each in second | (2x + 1)(x - 4) → 2x² - 7x - 4 | Write all terms before combining like terms |
| Factoring | Extract greatest common factor from all terms | 9x + 15xy → 3x(3 + 5y) | Factor completely; check by expanding to avoid errors |
Active recall
Quick check
1. Simplify: 4m + 7n - 2m + 3n
Combine like terms: 4m - 2m = 2m, 7n + 3n = 10n, so simplified expression is 2m + 10n.
2. Expand: 5y(3y - 2)
Multiply 5y by 3y = 15y², and 5y by -2 = -10y, so expanded expression is 15y² - 10y.
3. Expand: (x + 4)(x - 3)
Multiply x by x = x², x by -3 = -3x, 4 by x = 4x, 4 by -3 = -12; combine like terms -3x + 4x = x; final expression x² + x - 12.
4. Factor: 12a + 18ab
GCF is 6a; divide each term by 6a: 12a ÷ 6a = 2, 18ab ÷ 6a = 3b; factored form is 6a(2 + 3b).