Algebraic Operations
What this domain tests
The Algebraic operations domain tests whether you can rewrite expressions accurately and recognize structure before calculating. CLEP College Algebra items here emphasize polynomial arithmetic, factoring, rational expressions, exponent laws, radicals, and complex numbers. Most errors come from illegal cancellation or from ignoring restrictions on variables.
Polynomials and factoring
- Add and subtract by combining like terms; multiply each term of one polynomial by each term of the other. The degree of a product is the sum of the degrees.
- Memorize special products: $(a\pm b)^2=a^2\pm2ab+b^2$, $(a+b)(a-b)=a^2-b^2$, and $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$.
- Factoring strategy: greatest common factor first, then grouping, difference of squares, sum or difference of cubes, or trinomial trial. A polynomial is factored completely when no factor factors further over the integers.
Rational expressions and exponents
- Cancel factors, never terms: $\frac{x+2}{2}$ cannot become $x+1$.
- For $x^{m/n}$, take the $n$th root and raise to the $m$; negative exponents move the factor across the fraction bar.
- State excluded values where a denominator is zero before simplifying. Simplifying does not restore them.
Radicals and complex numbers
- Simplify radicals by extracting perfect-power factors; rationalize denominators using conjugates.
- Since $i^2=-1$, add and subtract complex numbers componentwise, multiply as binomials and replace $i^2$, and divide by multiplying numerator and denominator by the conjugate.
Worked example
Simplify $\frac{x^2-4}{x^2-x-6}\cdot\frac{x^2-9}{x+2}$ and list restrictions.
Factor: $\frac{(x-2)(x+2)}{(x-3)(x+2)}\cdot\frac{(x-3)(x+3)}{x+2}$. Before canceling, note $x\ne3$ and $x\ne-2$. Canceling $(x+2)$ and $(x-3)$ leaves $\frac{(x-2)(x+3)}{x+2}$, with $x\ne3,-2$. Check $x=0$: the original equals $\left(\frac{-4}{-6}\right)\left(\frac{-9}{2}\right)=-3$, and the simplified expression equals $\frac{(-2)(3)}{2}=-3$.
Traps that cost marks
- Writing $(a+b)^2=a^2+b^2$ and omitting the middle term $2ab$.
- Canceling a term shared by numerator and denominator rather than a common factor.
- Forgetting that a simplified rational expression keeps the original domain restrictions.
- Applying a negative exponent only to the coefficient, as in treating $(2x)^{-2}$ as $2x^{-2}$.