About this chapter
Algebraic methods opens Pure 3 with algebraic fractions, the skill that later chapters quietly depend on. Dividing an improper fraction before integrating it, simplifying a derivative, or rewriting a function to find its range all start here, and the June 2024 WMA13 paper used exactly this division as the first step of an integration question.
The chapter covers factorising and cancelling, multiplying, dividing, adding and subtracting algebraic fractions, solving equations that contain them, improper fractions, algebraic division by linear and quadratic divisors, and identities. The later questions ask why a graph is a straight line with points missing, and use the method of differences to sum a series.
The ten questions
- Q1 (4 marks): combine two algebraic fractions and show they simplify to (3x−5)/(2x−5).
- Q2 (5 marks): divide a quartic by x² + 3x to find A, B, C and D, then solve an equation.
- Q3 (6 marks): simplify three fractions to a single one, then show an equation has no real solutions.
- Q4 (5 marks): write an improper fraction as Ax + B + C/(2x+3), then solve an inequality with it.
- Q5 (5 marks): an equation with algebraic fractions, where one solution must be rejected with a reason.
- Q6 (9 marks): the factor theorem, then show f(x) = x + 2, and find the two points missing from the straight-line graph.
- Q7 (8 marks): divide a quartic by x² + 1 to find four constants, then an inequality using the remainder.
- Q8 (8 marks): a division of two fractions that simplifies to 1, the values where it is undefined, and an equation with no solutions.
- Q9 (9 marks): split 1/(r(r+1)) and sum a series by the method of differences, then the least n for a total over 0.99.
- Q10 (10 marks): an identity fixes k and C, then where the curve meets a line, and when it lies above y = 2x.
Key skills tested
Factorise and cancel. Factorise the numerator and denominator fully, then cancel common factors, never single terms. For example, (x² − 9)/(x² + 3x) = (x−3)/x.
Multiplying and dividing. To divide, multiply by the reciprocal, and factorise everything before cancelling.
Adding and subtracting. Use the lowest common denominator, found by factorising the denominators first.
Equations with fractions. Multiply through by the common denominator, then reject any value that makes a denominator zero.
Improper fractions. When the numerator's degree is at least the denominator's, divide first. For example, (3x² − 5x + 4)/(x−2) = 3x + 1 + 6/(x−2).
Algebraic division. Divide by (x + a) or by a quadratic. The remainder always has a lower degree than the divisor.
Identities. The identity sign means true for every x, so multiply out and compare coefficients, or substitute convenient values of x.
Showing results. Show every step: the common denominator, the expanded numerator and the factor that cancels.
Excluded values. Any x that makes a denominator zero is excluded, even if that factor later cancels.
Worked example
Question 4 from this chapter. f(x) = (6x² + 5x − 4)/(2x+3). (a) Write f(x) in the form Ax + B + C/(2x+3). (b) Hence find the set of values of x for which f(x) > 3x − 2.
(a) Divide by (2x+3). The first term is 3x, and 3x(2x+3) = 6x² + 9x, leaving −4x − 4. The next term is −2, and −2(2x+3) = −4x − 6, leaving a remainder of 2. So f(x) = 3x − 2 + 2/(2x+3).
(b) f(x) > 3x − 2 means 2/(2x+3) > 0, which happens when 2x + 3 is positive, so x > −3/2.
The mark scheme's note is worth remembering: the final mark in (a) needs f(x) written out in full, not just the values of A, B and C.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q2(a): an improper fraction written as a polynomial plus a proper fraction, with the function then stated in full, as in Q2, Q4 and Q10.
- June 2024, Q2(b): that divided form integrated straight after, which is where this chapter leads in Chapter 7.
Where marks are lost
- Stopping at the constants. In June 2024 it was common to find A, B and C and stop, when the function needed writing out in full for the final mark.
- Slips in the division. An incorrect x coefficient or constant in the quotient was the usual error in June 2024, and it carried into every later part.
- Cancelling terms. Only factors can cancel. Crossing out matching terms that are added or subtracted is never valid.
- Keeping an excluded value. A solution that makes a denominator zero must be rejected, and the reason stated.
Common questions
When do I need to divide before anything else?
Whenever the numerator's degree is equal to or higher than the denominator's. That is an improper fraction.
Division or comparing coefficients?
Both work. The June 2024 examiners saw both methods; long division was the most common, and either earns full marks.
Why is the graph in Q6 missing two points?
The simplified function x + 2 is undefined at the two values that made the original denominator zero, so those points are removed from the line.
Where this chapter leads
- Pure 3 Chapter 7: Integration: improper fractions divided before integrating, giving a log term.
- Pure 4 Chapter 2: Partial Fractions: splitting fractions into simpler parts.