Polynomials and Partial Fractions Notes — Sec 3 A-Math

Division algorithm

$P(x) \equiv D(x)\,Q(x) + R(x), \quad \deg R < \deg D$

P is the dividend, D the divisor, Q the quotient and R the remainder. Also deg Q = deg P − deg D. A linear divisor leaves a constant remainder; a quadratic divisor leaves a remainder of the form px + q.

Remainder Theorem

$\text{Remainder when } P(x) \text{ is divided by } (ax+b) = P\left(-\frac{b}{a}\right)$

Set the divisor equal to zero and solve for x. For a divisor (x − c), the remainder is P(c).

Factor Theorem

$(ax+b) \text{ is a factor of } P(x) \iff P\left(-\frac{b}{a}\right) = 0$

The special case of the Remainder Theorem where the remainder is zero. It works in both directions.

Sum and difference of cubes

a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²)

Signs follow SOAP: Same, Opposite, Always Positive. Take out any common factor first. The quadratic factor never has real roots.

Partial fractions: distinct linear factors

$\frac{px+q}{(ax+b)(cx+d)} \equiv \frac{A}{ax+b} + \frac{B}{cx+d}$

Each different linear factor gets one term with a constant numerator. A and B are constants to be found.

Partial fractions: repeated linear factor

$\frac{px^2+qx+r}{(ax+b)(cx+d)^2} \equiv \frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{(cx+d)^2}$

A squared factor needs two terms. The number of unknowns always equals the degree of the denominator.