Topic 1 of 6 · Form 5
Further Pure 1: roots, rational functions, series and induction
Sums and products of roots (Σα, Σαβ, αβγ); symmetric functions; Sₙ = αⁿ + βⁿ + γⁿ using recurrence from the equation; transformations (y = 2x, y = x + 1, y = x²).
Notes for this topic 1 file, in order
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Roots of polynomial equations
Sums and products of roots (Σα, Σαβ, αβγ); symmetric functions; Sₙ = αⁿ + βⁿ + γⁿ using recurrence from the equation; transformations (y = 2x, y = x + 1, y = x²).
Worked example: x³ − 2x² + 5x − 1 = 0 has roots α, β, γ. Find α² + β² + γ².
(Σα)² − 2Σαβ = 4 − 10 = −6
Rational functions
Vertical, horizontal and oblique asymptotes; intersections with axes; turning points; regions where the curve does not exist (using the discriminant); sketching y = |f(x)|, y = 1/f(x) and y² = f(x).
Summation of series
Standard results Σr, Σr², Σr³; method of differences; convergence and sums to infinity.
Worked example: Use the method of differences to find Σ 1/(r(r + 2)) from r = 1 to n.
1/(r(r + 2)) = ½[1/r − 1/(r + 2)] → ½[1 + ½ − 1/(n + 1) − 1/(n + 2)] = ¾ − (2n + 3)/(2(n + 1)(n + 2))
Proof by induction
Series, divisibility, matrix powers and derivative results.
Check yourself
- For 2x² − 3x + 5 = 0, find α + β and αβ.
- Find the oblique asymptote of y = (x² + 3x)/(x − 1).
- Evaluate Σr³ for r = 1 to 6.
Show answers
- 3/2 and 5/2
- y = x + 4 (since x² + 3x = (x − 1)(x + 4) + 4)
- 441
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