Algebra

• |x| is the size of x without its sign; the graph of y = |ax + b| is a V shape with vertex on the x-axis.
• |a| = |b| ⇔ a² = b², and |x − a| < b ⇔ a − b < x < a + b.
• Divide a polynomial of degree up to 4 by a linear or quadratic divisor to find quotient and remainder.
• Remainder theorem: the remainder on dividing p(x) by (ax − b) is p(b/a).
• Factor theorem: (ax − b) is a factor of p(x) if and only if p(b/a) = 0.

Distance from zero

|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 axis; the vertex is at x = −b/a.

When both sides are moduli, squaring is safe because both sides are non-negative. When one side is not a modulus, either split into cases (keeping each case's condition) or sketch both graphs and find where they cross.

p(x) = divisor × quotient + remainder

Long division works exactly as with numbers; write any missing powers with a zero coefficient. Dividing by a linear divisor leaves a constant remainder; dividing by a quadratic leaves a remainder of the form px + q.

The factor and remainder theorems let you skip the division: substitute the value that makes the divisor zero. Two pieces of information (two factors, or a factor and a remainder) give two equations for two unknown coefficients.