Polynomials & Quadratics

What Is a Polynomial? Terms, Degree and the Factor Theorem

A polynomial is a sum of terms of the form $ax^n$, where each power $n$ is a whole number, such as $4x^3 - x + 7$. Its degree is the highest power present.

How to Factorise Quadratics: Sum-Product Method and Difference of Two Squares

Factorising a quadratic rewrites it as a product of brackets: $x^2 + 7x + 12 = (x + 3)(x + 4)$. For $x^2 + bx + c$, look for two numbers that add to $b$ and multiply to $c$.

Completing the Square and Vertex Form of a Quadratic

Completing the square rewrites $x^2 + bx + c$ as $(x + \frac{b}{2})^2 + k$. The result, vertex form $a(x - h)^2 + k$, shows the parabola's turning point $(h, k)$ directly.

The Quadratic Formula and the Discriminant Explained

The quadratic formula solves any equation $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant $b^2 - 4ac$ tells you how many real solutions there are.

Rational Functions: Asymptotes, Holes and Algebraic Fractions

A rational function is one polynomial divided by another, like $f(x) = \frac{1}{x - 2}$. Its graph can have asymptotes, lines it approaches but never reaches, and holes, single missing points.

Rationalising the Denominator: Surds and Conjugates

Rationalising the denominator rewrites a fraction so no root is left on the bottom, without changing its value. Multiply top and bottom by the root, or by the conjugate when the bottom is a sum like $3 + \sqrt2$.