Numbers and Algebraic Expressions
This domain covers integer and decimal arithmetic, square roots, and the manipulation of algebraic expressions built from variables. You should be fluent in the order of operations, sign rules for negative numbers, and the distributive law, since almost every later topic leans on them.
Core skills
- Combine like terms and simplify expressions with parentheses, remembering that a minus sign in front of parentheses flips every inner sign.
- Expand products such as (a+b)(a−b)=a²−b² and (a±b)²=a²±2ab+b².
- Simplify square roots: pull out perfect squares (√48=4√3) and rationalize denominators.
- Evaluate expressions by substitution, keeping careful track of negative signs.
Common traps
A frequent error is dropping a sign when subtracting a bracket: 5−(x−3) becomes 5−x+3, not 5−x−3. Another is treating √(a+b) as √a+√b, which is false. When substituting a negative value, always bracket it first: x² with x=−3 is (−3)²=9, not −9.
Worked example
Simplify 3(2x−4)−2(x−5). First expand: 6x−12−2x+10. Then collect: 4x−2. Check by substituting x=1 in the original and the result: original gives 3(−2)−2(−4)=−6+8=2; result gives 4−2=2. Matching values confirm the simplification.
Rule of thumb: expand fully, then collect, then spot-check with a simple number.