Algebra
What this domain covers
Algebra in NSSCAS Mathematics 8227 rewards exact manipulation rather than decimal approximation. Expect simplification of surds and rationalising denominators, index and logarithm laws applied within one expression, completing the square for a vertex or range, the discriminant for classifying roots, the factor and remainder theorems for cubics, linear and quadratic inequalities, and binomial expansion of (a + b)^n.
Core rules
- Indices: a^m · a^n = a^(m+n); a^(m/n) = (n√a)^m; a^(−n) = 1/a^n; a^0 = 1.
- Surds: √(ab) = √a · √b; rationalise 1/(a + √b) by multiplying numerator and denominator by the conjugate a − √b.
- Logarithms: log_a x = n means a^n = x; log_a(xy) = log_a x + log_a y; log_a(x^k) = k log_a x; change of base log_a x = log_b x ÷ log_b a.
- Quadratic ax² + bx + c: the discriminant b² − 4ac is positive, zero or negative according to whether there are two, one or no real roots.
- Remainder theorem: dividing f(x) by (x − a) leaves f(a); f(a) = 0 makes (x − a) a factor.
- Binomial: (a + b)^n = Σ nCr · a^(n−r) · b^r, with the term containing x^r obtained from nCr when the second term is x.
Worked example
Solve 3^(2x) − 10·3^x + 9 = 0. Let y = 3^x. Then y² − 10y + 9 = 0, so (y − 1)(y − 9) = 0. Hence 3^x = 1 giving x = 0, or 3^x = 9 giving x = 2. Both are valid because 3^x > 0 for every real x, so neither root is rejected.
Traps
- Writing log(x + y) = log x + log y; the product law applies only to a product.
- Solving a quadratic inequality without a sign diagram and reporting the interval between the roots when the solution lies outside them.
- Using nCr with the wrong r: in (1 + kx)^n the coefficient of x^r comes from nCr(k)^r.
- Forgetting that a logarithm's argument must be positive when the base is unchanged by the manipulation.