Basic Algebra
Basic Algebra examines quadratic expressions and their roots, the remainder and factor theorems for polynomials, resolution into partial fractions, and the laws of exponents and logarithms. The questions are usually short but very sensitive to sign slips.
Quadratics and polynomials
For ax² + bx + c with a ≠ 0, the discriminant Δ = b² − 4ac decides the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives equal real roots, Δ < 0 gives a conjugate pair of complex roots. The sum of the roots is −b/a and the product is c/a. The remainder theorem says the remainder on dividing p(x) by (x − a) is p(a); the factor theorem says (x − a) is a factor exactly when p(a) = 0.
Partial fractions
First check that the degree of the numerator is smaller than the degree of the denominator; if not, divide. Distinct linear factors each receive a constant numerator, a repeated factor receives one term for every power, and an irreducible quadratic receives a linear numerator.
Exponents and logarithms
aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, log(ab) = log a + log b, log(a/b) = log a − log b, log aⁿ = n log a, and log_b a = log a / log b. A logarithm needs a positive argument, a base greater than zero, and a base different from 1.
Worked example
Split 1/((x+1)(x+2)) as A/(x+1) + B/(x+2). Then 1 = A(x+2) + B(x+1). Putting x = −1 gives A = 1; putting x = −2 gives B = −1. So the answer is 1/(x+1) − 1/(x+2).
Traps
- Applying root formulas when a could be zero.
- Skipping the division step for an improper fraction.
- Retaining solutions that make a logarithm argument negative.