Algebra

• |x| is the distance of x from 0. The graph of y = |ax + b| is a V with its vertex on the x-axis at x = −b/a.
• Key relations: |a| = |b| ⇔ a² = b², and |x − a| < b ⇔ a − b < x < a + b.
• Polynomial division (degree up to 4, by a linear or quadratic divisor) gives a quotient and a remainder.
• Factor theorem: (ax + b) is a factor of p(x) ⇔ p(−b/a) = 0. Remainder theorem: the remainder on dividing by (ax + b) is p(−b/a).
• Partial fractions for denominators (ax + b)(cx + d)(ex + f), (ax + b)(cx + d)² and (ax + b)(cx² + d).
• (1 + x)ⁿ for rational n is an infinite series, valid only for |x| < 1.

Squaring, cases and sketches

|x| = x when x ≥ 0 and |x| = −x when x < 0. To sketch y = |ax + b|, draw y = ax + b and reflect the part below the x-axis in the x-axis, giving a V shape.

When both sides are moduli, square: both sides are non-negative so squaring keeps the equation or inequality equivalent. When only one side is a modulus, either split into cases (keeping each case's condition) or sketch both graphs and find where they cross.

Quotient, remainder and factors

Dividing p(x) by a divisor d(x) gives p(x) = d(x) × q(x) + r(x), where the remainder r(x) has lower degree than d(x). A linear divisor leaves a constant remainder; a quadratic divisor leaves a remainder of the form Ax + B. Use long division, or compare coefficients in the identity.

The factor and remainder theorems let you find remainders without dividing, and find unknown coefficients by forming simultaneous equations. The divisor may have a coefficient of x that is not 1, such as (2x + 1): substitute x = −½.

Choosing the right form

A proper rational function (numerator degree lower than denominator degree) can be split into simpler fractions. Choose the form from the denominator, then multiply through and find the constants by substituting convenient values of x and comparing coefficients.