Algebra & functions

Rational indices, surds, quadratics and the discriminant, simultaneous equations, inequalities, polynomial division and the factor theorem.

11 min read

Mathematics 2 builds on the algebra of Maths 1 and adds the tools used throughout the rest of the module: completing the square, the discriminant, polynomial division, the factor and remainder theorems, and the language of functions.

The index laws hold for all rational exponents: $a^{m / n} = n a^{m} = (n a)^{m}$ , with $a^{- m} = \frac{1}{a ^{m}}$ and $a^{0} = 1$ (for $a > 0$ ).

Simplify surds with $ab = a b$ and rationalise denominators using the conjugate. The convention is that $a$ means the positive root.

Worked example

Simplify $\frac{5}{3 + 2 5}$ .

Multiply by the conjugate $\frac{3 - 2 5}{3 - 2 5}$ . Denominator: $(3)^{2} - (25)^{2} = 9 - 20 = - 11$ .

$\frac{5 ( 3 - 2 5 )}{- 11} = \frac{3 5 - 10}{- 11} = \frac{10 - 3 5}{11} .$

A quadratic $a x^{2} + b x + c$ can be solved by factorising, by the formula $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ , or by completing the square.

The discriminant $Δ = b^{2} - 4 a c$ controls the roots: $Δ > 0$ two real roots, $Δ = 0$ one repeated root, $Δ < 0$ no real roots. This is a favourite ESAT tool for "for what values of $k$ …" problems.

Worked example

For what values of $k$ does $x^{2} + k x + 9 = 0$ have equal roots?

Equal roots ⇒ $Δ = 0$ : $k^{2} - 4 (1) (9) = 0 \Rightarrow k^{2} = 36 \Rightarrow k = \pm 6.$

Worked example

Complete the square for $2 x^{2} - 8 x + 5$ .

Factor out $2$ : $2 (x^{2} - 4 x) + 5 = 2 [(x - 2)^{2} - 4] + 5 = 2 (x - 2)^{2} - 3.$ Minimum value $- 3$ at $x = 2$ .

Solve a linear + quadratic pair by substitution: rearrange the linear equation and substitute into the quadratic, giving a single quadratic to solve.

Worked example

Solve $y = x + 1$ and $x^{2} + y^{2} = 25$ .

Substitute: $x^{2} + (x + 1)^{2} = 25 \Rightarrow 2 x^{2} + 2 x - 24 = 0 \Rightarrow x^{2} + x - 12 = 0 \Rightarrow (x + 4) (x - 3) = 0.$

So $x = - 4, y = - 3$ or $x = 3, y = 4$ , the two points where the line meets the circle.

Solve linear inequalities like equations (flip the sign when multiplying/dividing by a negative). For a quadratic inequality, find the roots, sketch the parabola, and read off where it is above or below the axis.

Worked example

Solve $x^{2} - x - 6 > 0$ .

Roots of $x^{2} - x - 6 = (x - 3) (x + 2)$ are $x = 3, - 2$ . The upward parabola is positive outside the roots:

$x < - 2 or x > 3.$

Exam tip

For a " $> 0$ " with an upward parabola, the answer is the outside region; for " $< 0$ ", it is between the roots. A quick sketch prevents the most common sign error here.

Polynomials can be divided by a linear $a x + b$ or a quadratic $a x^{2} + b x + c$ (algebraic long division), leaving a quotient and remainder.

Remainder theorem: the remainder when $f (x)$ is divided by $(x - a)$ is $f (a)$ . Factor theorem: $(x - a)$ is a factor exactly when $f (a) = 0$ . Use it to find one root, then factorise what remains.

Worked example

Show $(x - 2)$ is a factor of $f (x) = x^{3} - 3 x^{2} + 4$ , and factorise fully.

$f (2) = 8 - 12 + 4 = 0$ , so $(x - 2)$ is a factor. Dividing gives $x^{3} - 3 x^{2} + 4 = (x - 2) (x^{2} - x - 2) = (x - 2) (x - 2) (x + 1) = (x - 2)^{2} (x + 1) .$

Graph of y equals the modulus of x, a V-shape with its vertex at the origin, never going below the x-axis. — y = |x| takes the size of x and ignores its sign, so it is never negative.

A function maps each input to exactly one output; it may be many-to-one (like $f (x) = x^{2}$ ) or one-to-one. Two functions to know precisely: $f (x) = x$ always means the positive square root, and $f (x) = ∣ x ∣$ (the modulus) gives the size of $x$ with its sign removed, so $∣ - 3 ∣ = 3$ .

Treating $x^{2}$ as $x$ rather than $∣ x ∣$ .
Getting the region wrong for a quadratic inequality (always sketch).
Forgetting the " $\pm$ " when a discriminant or square root gives two values.

Algebra & functions

Overview

Indices & surds

Quadratics, the discriminant & completing the square

Simultaneous equations

Linear & quadratic inequalities

Polynomials, division & the factor theorem

Functions, √x and |x|

Common mistakes