Chapter 2: Polynomials

Algebra is built on three ideas: variables, operations, and patterns. A polynomial is the simplest object that combines all three , it is an algebraic expression made by adding and multiplying a variable with constants, like $x^2 - 5x + 6$ or $3x^3 - 2x + 7$. This chapter is your first careful tour of polynomials: what they are, how to operate on them, and the theorems that turn them into a powerful tool.

The heart of the chapter is the connection between zeros and factors. A number $a$ is a zero of $p(x)$ if $p(a) = 0$. The Factor Theorem says: $a$ is a zero precisely when $(x - a)$ is a factor of $p(x)$. This deceptively simple result lets you factorise cubics , which would otherwise look impossible , by guessing one zero and dividing.

Equally important are the algebraic identities: $(a+b)^2$, $(a-b)^2$, $a^2 - b^2$, $(a+b)^3$, $(a-b)^3$, $a^3 \pm b^3$, and $a^3 + b^3 + c^3 - 3abc$. These are short formulas that turn complicated expansions into one-line answers. They are not optional decorations; they are the alphabet of all later algebra. You will use $(a+b)^2 = a^2 + 2ab + b^2$ when you solve quadratics, when you compute distances, when you prove inequalities, and when you complete the square.

Polynomials show up everywhere in mathematics and the sciences. The motion of a falling stone is described by a quadratic. Profits and costs in economics are polynomial. Computers approximate trigonometric functions, exponentials, and logarithms by polynomial fragments. Every smooth curve, on a small enough scale, is well-approximated by a polynomial. So the algebraic gymnastics in this chapter is not arbitrary , it is the toolkit you need for the rest of school maths and beyond.

The chapter is short on theorems but long on practice. The goal is fluency: by the end you should see the factorisation of $x^3 - 7x + 6$ in your head, recognise that $a^2 + b^2 + c^2 - ab - bc - ca = \tfrac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2]$, and never panic at a cubic divided by a linear.

What's inside

• Polynomials in one variable , definitions, degree, types (linear, quadratic, cubic).
• Zeros of a polynomial , what they are, how to find them, how many a polynomial of given degree can have.
• Remainder Theorem , the remainder of $p(x) \div (x - a)$ is $p(a)$.
• Factor Theorem , $(x - a)$ is a factor of $p(x)$ iff $p(a) = 0$.
• Algebraic identities , square and cube identities, $a^3 \pm b^3$, $a^3 + b^3 + c^3 - 3abc$.
• Factorisation , splitting the middle term in quadratics, and using the Factor Theorem for cubics.

Key results / Formula card

Memorise this card. Nine out of ten questions in the chapter are direct applications.