Before Calculus: 10 Algebra Skills You Should Know

1. Work confidently with fractions

Fractions represent division. When adding them, rewrite each fraction using a common denominator. When multiplying, multiply numerators and denominators. Keep intermediate fractions exact rather than rounding too early.

Common mistake. Adding denominators: $\frac{1}{2}+\frac{1}{3}$ is not $\frac{2}{5}$. You need equal-sized pieces before adding their counts.

2. Expand without losing signs

The distributive property applies to every term inside parentheses. A negative sign outside parentheses means multiplication by $-1$, so it changes every sign inside.

Common mistake. Distributing only to the first term, or turning $-2(x+4)$ into $-2x+8$.

3. Recognize useful factorizations

Factoring rewrites a sum as a product. Check for a common factor first, then patterns such as a difference of squares or a quadratic trinomial. Expand your answer to verify it.

Common mistake. The product $(a+b)(a-b)$ equals $a^2-b^2$, not $a^2+b^2$.

4. Solve linear equations with balanced operations

An equation stays equivalent when you perform the same valid operation on both sides. Clear parentheses, collect terms, then divide by the nonzero coefficient of the unknown.

Common mistake. Moving a term across the equals sign without understanding the addition or subtraction being performed.

5. Keep every valid quadratic solution

A quadratic may have two distinct real solutions, one repeated solution, or no real solutions. Factoring is useful when available; the quadratic formula works more generally.

Common mistake. Dividing an equation by $x$ can lose the solution $x=0$. Check that case before dividing.

6. Use exponent and radical rules accurately

For a nonzero base, negative exponents mean reciprocals. Multiplying powers with the same base adds exponents; raising a power to a power multiplies exponents.

Common mistake. For real $x$, $\sqrt{x^2}=|x|$, not always $x$. A square root returns a nonnegative value.

7. Preserve restrictions when cancelling factors

You may cancel common factors, not individual terms within a sum. Record excluded values from the original denominator before simplifying.

Common mistake. Cancelling the $x$ terms in $\frac{x+2}{x}$ is invalid; the numerator is a sum, not a product with factor $x$.

8. Understand functions, composition and domains

A function assigns an output to each permitted input. Its domain is the set of inputs for which the defining expression makes sense. Composition means substitution, not multiplication.

Common mistake. $f(x+1)$ is not generally $f(x)+1$. Replace every occurrence of the input before simplifying.

9. Connect exponentials and logarithms

Logarithms undo exponentials. For a positive base $a\ne1$, $\log_a(b)=c$ means $a^c=b$. In real-valued work, the logarithm’s argument must be positive.

Common mistake. There is no rule $\ln(a+b)=\ln a+\ln b$. For positive $a$ and $b$, it is the product rule $\ln(ab)=\ln a+\ln b$.

10. Calculate slopes and average rates

A slope compares a change in output with a change in input. Keep the subtraction order consistent. Rates also have units: metres divided by seconds gives metres per second.

Common mistake. Dividing output values instead of output differences, or reversing only one subtraction.

Your next revision session

• Mark each skill comfortable, needs a reminder, or needs practice. These are study labels, not a prediction of your grade.
• Start with fractions, signs and equation solving if those are uncertain: they appear throughout the other skills.
• Explain your steps aloud, verify by substitution or expansion, and retry missed questions later without notes.

Connect these skills to your course