What you will learn in Polynomials & Quadratics

1. 1. Polynomials, Degree and Shape

   Degree decides where both ends of the curve go, and puts a ceiling on how many times it can turn.

2. 2. Remainder and Factor Theorems

   Divide by x minus c, or substitute c. Two routes, one number — and when it is zero you have found a factor.

3. 3. Factorising Polynomials

   The rational-root theorem turns an infinite guess into a finite list you can finish.

4. 4. Zeros, Multiplicity and Graph Behaviour

   Why a repeated root touches the axis rather than crossing it, and why a triple root flattens on the way through.

5. 5. Three Forms of a Quadratic

   One curve, three ways of writing it, and a different question answered at a glance by each.

6. 6. Completing the Square

   The name is literal: there is a square, and a corner missing from it.

7. 7. Deriving and Using the Quadratic Formula

   Completing the square on the general form, one step at a time, until the formula falls out.

8. 8. The Discriminant and the Nature of Roots

   One number decides whether the curve cuts the axis, touches it, or misses it entirely.

9. 9. Sum and Product of Roots

   The sum and product of the roots are visible in the coefficients before any root is found.

10. 10. Quadratics in the Real World

    A trajectory, a fence and a price. Three problems, one curve, and one bridge into Physics.

11. 11. Rational Expressions and Excluded Values

    Cancelling changes how an expression is written, never where it is defined.