3.2 Algebraic expressions: simplifying, distributing & combining like terms
Key Takeaways
- The distributive property a(b + c) = ab + ac multiplies the outside factor by every inside term; watch signs with a negative multiplier.
- Combine like terms by adding the coefficients of terms with identical variable parts; 5a and 5b are not like terms.
- Order of operations is PEMDAS, but multiply/divide share one rank and add/subtract share one rank — work each pair left to right.
- Evaluate an expression by substituting values in parentheses, then applying the order of operations to protect negative signs.
- Commutative and associative properties let you reorder and regroup addition and multiplication, but never subtraction or division.
- Function notation f(x) means substitute the input for x: for f(x)=2x+1, f(3)=7.
Building and simplifying expressions
An algebraic expression is a combination of numbers, variables, and operations — for example 3(2x + 5) - 2(4x - 3). Unlike an equation, an expression has no equals sign, so you do not solve it; you simplify it into the cleanest equivalent form. Three tools do almost all of the work on the PERT: the distributive property, combining like terms, and the order of operations. A fourth skill, evaluating an expression by substitution, turns a general expression into a single number.
The distributive property
The distributive property says a(b + c) = ab + ac: multiply the outside factor by every term inside the parentheses. Watch the signs carefully, especially with a negative multiplier.
3(2x + 5) = 3·2x + 3·5 = 6x + 15-2(4x - 3) = -2·4x + (-2)(-3) = -8x + 6
A negative distributed across a subtraction flips that term to positive, which is where most careless errors happen. Slow down and multiply each inside term separately.
Combining like terms
Like terms have the exact same variable part — the same letters raised to the same exponents. You may add or subtract their coefficients; the variable part stays unchanged. So 5a and -2a are like terms, 5a and 5b are not, and x and x² are not.
Simplify 5a + 3b - 2a + 7b - 4. Group like terms: (5a - 2a) + (3b + 7b) - 4 = 3a + 10b - 4. The constant -4 has no variable partner, so it simply stays.
Now combine both tools on 3(2x + 5) - 2(4x - 3):
- Distribute:
6x + 15 - 8x + 6. - Combine like terms:
(6x - 8x) + (15 + 6) = -2x + 21.
The simplified expression is -2x + 21.
Try a longer one: simplify 4(x - 2) + 3(2x + 1) - 5. Distribute each group: 4x - 8 + 6x + 3 - 5. Combine like terms: (4x + 6x) + (-8 + 3 - 5) = 10x - 10. Notice how the three constants combine to -10; if a fully factored form is requested, 10x - 10 can be written 10(x - 1).
Order of operations (PEMDAS)
When an expression mixes operations, you must evaluate them in a fixed order. A common reminder is PEMDAS.
| Step | Operation | Note |
|---|---|---|
| P | Parentheses / grouping | Innermost first |
| E | Exponents | Powers and roots |
| MD | Multiply & Divide | Left to right, equal priority |
| AS | Add & Subtract | Left to right, equal priority |
The biggest trap: multiplication and division share one rank (do them left to right in the order they appear), and so do addition and subtraction. PEMDAS is not six separate steps.
Evaluate 4 + 3 × 2² - (6 - 2):
- Parentheses:
6 - 2 = 4, giving4 + 3 × 2² - 4. - Exponent:
2² = 4, giving4 + 3 × 4 - 4. - Multiply:
3 × 4 = 12, giving4 + 12 - 4. - Add and subtract left to right:
4 + 12 = 16, then16 - 4 = 12.
The value is 12.
Evaluating by substitution
To evaluate an expression, replace each variable with its given value and then follow the order of operations. Use parentheses around every substituted value to protect signs.
Evaluate 2x² - 3y for x = 3 and y = 4: 2(3)² - 3(4) = 2·9 - 12 = 18 - 12 = 6.
Evaluate 5 - 2x for x = -3: 5 - 2(-3) = 5 + 6 = 11. Notice how -2 times -3 becomes +6 — the parentheses keep you from mistakenly writing 5 - 2 - 3.
When an expression has a fraction bar or nested grouping, treat the numerator and denominator (or the innermost parentheses) as their own grouped calculations before combining. For example, evaluate (2x + 4) / 2 for x = 5: the numerator is 2(5) + 4 = 14, then 14 / 2 = 7. Grouping symbols always resolve first, even when they are written as a division bar rather than round brackets.
Properties that let you rearrange
Two properties justify the regrouping you do while simplifying:
- Commutative property — order does not matter for addition or multiplication:
a + b = b + aandab = ba. This is why3b + 5acan be rewritten as5a + 3b. - Associative property — grouping does not matter for addition or multiplication:
(a + b) + c = a + (b + c)and(ab)c = a(bc). This is why you can add a column of like terms in any grouping.
Neither property applies to subtraction or division: 8 - 3 ≠ 3 - 8. When you "move" a subtracted term, first rewrite it as adding a negative so the commutative property applies safely.
Quick reminder
Simplify in this order: distribute to clear parentheses, combine like terms, and keep the order of operations in mind for any arithmetic along the way. To evaluate, substitute with parentheses, then run PEMDAS. Master these routines and the algebra portion of the PERT becomes steady bookkeeping rather than guesswork.
Function Notation
Function notation writes a rule as f(x) ("f of x"), where x is the input. Evaluating a function means substituting a value for x and simplifying — the same skill as evaluating an expression. For example, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7, and f(−4) = 2(−4) + 1 = −7. The notation g(x), h(x), etc. work the same way; f(a) is just "the output when the input is a." Watch the order of operations: for f(x) = x² − 3x, f(5) = 25 − 15 = 10, not (5)² − 3 first. Function notation appears on the PERT when a rule is given and you are asked for the value at a specific input.
Simplify 3(2x + 5) - 2(4x - 3).
Using the order of operations, evaluate 4 + 3 × 2² - (6 - 2).
Evaluate 5 - 2x when x = -3.