Pure 2: Algebraic Methods

This lesson focuses on using proof by contradiction to demonstrate that a given statement must be true by assuming the opposite and reaching a logical contradiction.

Teacher’s tip:

Start by assuming the opposite of what you want to prove. Use known properties (e.g. parity of integers, rationality, divisibility) to reach a contradiction. For example:

• Assume √2 is rational, then show it leads to a contradiction in lowest terms
• Use properties of even and odd numbers to show contradictions in expressions Always state your contradiction clearly and conclude that the original statement must therefore be true.

This lesson focuses on simplifying algebraic fractions, including multiplying and dividing rational expressions by factorising and cancelling common factors.

Teacher’s tip:

Factorise all numerators and denominators fully before simplifying.

• When dividing, flip the second fraction and multiply
• Cancel any common linear or quadratic factors Always check for restrictions on variables (e.g. values that make a denominator zero), and write simplified answers with clear working.

This lesson focuses on expressing algebraic fractions as partial fractions, especially when the denominator factorises into linear terms.

Teacher’s tip:

Factor the denominator completely.

For distinct linear factors:

\frac{P(x)}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}

Multiply through by the denominator to eliminate fractions

Solve for unknowns by substituting values or equating coefficients For repeated or quadratic factors, extend the form accordingly. Always simplify the final expression and check by expanding back.

This lesson focuses on partial fractions with repeated linear factors, where the denominator includes terms like (x – a)².

Teacher’s tip:

For repeated linear factors, include terms for each power:

\frac{A}{x - a} + \frac{B}{(x - a)^2}

Multiply through by the full denominator, simplify, and solve for constants by substitution or comparing coefficients. Always check by expanding the final result to ensure it matches the original expression.

This lesson focuses on simplifying improper algebraic fractions, where the degree of the numerator is greater than or equal to the denominator, and then expressing them as partial fractions.

Teacher’s tip:

Start by using algebraic division to rewrite the improper fraction in the form:

\text{Polynomial} + \frac{\text{Remainder}}{\text{Denominator}}

Then decompose the remaining proper fraction into partial fractions. Always check your result by multiplying back to confirm it matches the original expression.