Polynomial functions — A/L Combined Mathematics

What must I remember about Polynomial functions?

• A polynomial function of degree n is f(x) = an*xⁿ + a(n-1)*x^n-1 + ... + a1*x + a0 where an ≠ 0 and coefficients are real constants.
• Division Algorithm: For polynomials f(x) and divisor g(x) ≠ 0, there exist unique polynomials q(x) [quotient] and r(x) [remainder] such that f(x) = g(x)*q(x) + r(x), where deg(r) < deg(g) or r(x) = 0.
• Remainder Theorem: When a polynomial f(x) is divided by a linear factor (ax - b), the remainder is the constant value f(b/a).
• Factor Theorem: A linear polynomial (ax - b) is a factor of f(x) if and only if f(b/a) = 0.
• Polynomial Identity Principle: Two polynomials of degree ≤ n that are equal for at least (n + 1) distinct values of x are identical for all x, meaning corresponding coefficients must be equal.

Test yourself (1 minute)

1. Repeated factor condition

   (x - a)² is a factor of f(x) if and only if f(a) = 0 AND f'(a) = 0.

2. Quadratic divisor remainder

   When f(x) is divided by a quadratic divisor (x - a)(x - b), the remainder is linear: r(x) = Ax + B; find A and B by substituting x = a and x = b into f(x) = (x - a)(x - b)q(x) + (Ax + B).

3. Rational Root Theorem

   If a polynomial with integer coefficients has a rational root p/q (in lowest terms), then p must divide the constant term a0, and q must divide the leading coefficient an.

Practise Polynomial functions questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 4

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Year 1 · Term 1