Simplifying changes the form of an expression without changing its value. The goal is an equivalent expression with every distribution completed and every possible group of like terms combined.
Terms, coefficients, and like terms
- Term
- A part separated by addition or subtraction. In \(4x^2-3x+7\), the terms are \(4x^2\), \(-3x\), and \(7\).
- Coefficient
- The signed numerical factor of a variable term. The coefficient of \(x\) in \(-5x\) is \(-5\).
- Like terms
- Terms with identical variable parts, including exponents. \(3x^2\) and \(-7x^2\) are like; \(3x^2\) and \(3x\) are unlike.
Three properties that organize simplification
| Property | General form | What changes | Original example |
|---|---|---|---|
| Commutative | \(a+b=b+a\) | Order changes | \(3x+8-5x=3x-5x+8\) |
| Associative | \((a+b)+c=a+(b+c)\) | Grouping changes | \((x+4)+6)=x+(4+6)\) |
| Distributive | \(a(b+c)=ab+ac\) | A factor multiplies every term | \(-2(x-5)=-2x+10\) |
A dependable simplification workflow
- Distribute
Remove grouping by multiplying each internal term by the complete signed factor outside.
- Reorder
Move each term with its sign so like variable parts are adjacent.
- Combine
Add or subtract coefficients of like terms; combine constants separately.
- Audit
Check that no grouping remains to distribute and no like terms remain to combine.
Negative and fractional coefficients
Simplify \(\frac12(6x+8)-\frac13(3x-9)\).
- Interpret
Distribute the fractional coefficients: \(3x+4-(x-3)\).
- Set up
The second result is subtracted, so distribute \(-1\): \(3x+4-x+3\).
- Work
Combine like terms: \(3x-x=2x\) and \(4+3=7\).
- Check
Substitution with a test value, such as \(x=2\), gives \(11\) in both forms.
Several variable parts
Simplify \(5x^2-2x+3-3x^2+7x-8\).
- Interpret
Separate the \(x^2\), \(x\), and constant groups.
- Set up
Write \((5-3)x^2+(-2+7)x+(3-8)\).
- Work
Calculate the coefficients to obtain \(2x^2+5x-5\).
Check your understanding
Simplify \(-3(2x-4)+5x-2\).
- \(-x+10\)
- \(-x-14\)
- \(11x+10\)
- \(-x-10\)
Show answer and explanation
Answer: \(-x+10\)
Distribute to get \(-6x+12+5x-2\), then combine to obtain \(-x+10\).
Key takeaways
What to remember
- Like terms must have identical variable parts and exponents.
- A term carries its sign when it moves.
- Distribute a signed factor to every term inside grouping.
- For fractional coefficients, use common denominators and simplify each product before combining.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.