Polynomials and Rational Expressions
The exam checks zeros of polynomials, the link between zeros and coefficients, the division algorithm, and simplification of rational expressions with stated restrictions.
Core rules
- A zero of p(x) is a value c with p(c) = 0; graphically these are the x-intercepts.
- For ax² + bx + c: sum of zeros = −b/a, product = c/a.
- For ax³ + bx² + cx + d: sum = −b/a, sum of products taken two at a time = c/a, product = −d/a.
- Division algorithm: p(x) = g(x)q(x) + r(x), where degree r < degree g. If r = 0, g is a factor.
- Remainder theorem: the remainder on dividing p(x) by (x − a) is p(a); p(a) = 0 means (x − a) is a factor.
- A rational expression is simplified by factorising numerator and denominator and cancelling common factors, recording that cancelled factors must not be zero.
Worked example
Find the zeros of 2x² − 7x + 3 and verify the coefficient relations. Split the middle term: 2x² − 6x − x + 3 = 2x(x − 3) − 1(x − 3) = (2x − 1)(x − 3). Zeros are 1/2 and 3. Check: sum = 1/2 + 3 = 7/2 = −(−7)/2 and product = 3/2, matching c/a.
Traps
Dropping the minus sign in −b/a; treating a zero as a coefficient; cancelling terms instead of factors in a rational expression; forgetting degree of the zero polynomial is undefined; not excluding values that make a cancelled denominator zero.