FOUNDATIONS 06
Algebraic Manipulation
Algebraic manipulation changes the form of an expression without changing its mathematical value.
The main idea
The same quantity can often be written in several equivalent forms. The best form depends on what we want to do next.
\[\boxed{x^2-5x+6=(x-2)(x-3)}\]The expanded form shows the individual terms; the factorised form immediately reveals the roots \(x=2\) and \(x=3\). Algebraic manipulation is therefore not merely tidying notation: it exposes useful mathematical structure.
1. Simplifying like terms
Only terms with exactly the same variable part can be combined. Their coefficients are added or subtracted.
\[3x+5x=(3+5)x=8x.\]But \(3x+5x^2\) cannot be combined because \(x\) and \(x^2\) are different powers.
\(=(4-2)x^2+(3+5)x\)
\(=2x^2+8x\)
2. Expanding a bracket
The distributive law says that a factor outside a bracket multiplies every term inside:
\[a(b+c)=ab+ac.\]\(=3\cdot x+3\cdot4\)
\(=3x+12\)
3. Expanding two brackets
Every term in the first bracket must multiply every term in the second.
\(=x^2+3x+2x+6\)
\(=x^2+5x+6\)
4. Factorising reverses expansion
Factorising rewrites a sum as a product. First look for a common factor.
\[6x+9=3(2x+3).\]For quadratics, factorisation can reveal values that make the expression zero.
\(=(x+2)(x+3)\)
because \(2+3=5\) and \(2\cdot3=6\)
5. Rearranging formulas
We often need to make a different variable the subject. The same equality-preserving rules used for equations apply.
If \(y=3x+5\), isolate \(x\).
\(y-5=3x\)
\(\boxed{x=\frac{y-5}{3}}\)
6. Algebraic fractions
Algebraic fractions obey the same principles as numerical fractions. To add fractions, use a common denominator.
\(=\frac{y}{xy}+\frac{x}{xy}\)
\(=\frac{x+y}{xy}\), \(x,y\ne0\)
7. Cancelling means cancelling factors
Cancellation is division by a common non-zero factor. It does not work across addition or subtraction.
\(x\) is a factor of the whole numerator.
8. Simplifying rational expressions
Factor first, then cancel common factors while retaining restrictions from the original expression.
\(=\frac{(x-2)(x+2)}{x-2}\)
\(=x+2,\qquad x\ne2\)
The simplified formula is \(x+2\), but the original expression was undefined at \(x=2\).
9. Fractions inside fractions
A complex fraction can often be simplified by multiplying the numerator and denominator by a common denominator.
\(=\frac1x\cdot\frac y1\)
\(=\frac yx\), \(x,y\ne0\)
10. Choosing the useful form
Equivalent forms emphasize different information. Good algebra means choosing the form that makes the next step easiest.
Algebraic manipulation preserves mathematical meaning while changing form. Expand to expose terms, factorise to expose factors and roots, rearrange to isolate a variable, and cancel only common non-zero factors. Always preserve restrictions such as excluded denominator values.
Why this matters for calculus
Many calculus problems become simple only after algebraic manipulation. Difference quotients are often factorised before taking a limit; derivatives may need simplification; integration may require expansion, factorisation or algebraic fractions; and equivalent formulas can reveal zeros, signs and singularities that are hidden in another form.