Further algebra and functions

Welcome to Further Algebra and Functions!

In this chapter, we are going to take the algebra skills you've mastered in A-level Maths and supercharge them. We will explore how the roots of equations relate to their coefficients, learn how to approximate complex functions using simple polynomials (Maclaurin series), and dive into the world of sketching complex graphs like hyperbolas and ellipses.

Why is this important? These tools are used everywhere from physics (calculating orbits of planets) to engineering (designing bridges) and computer science (optimizing algorithms). Don't worry if some of this feels a bit abstract at first—we'll take it one step at a time!

1. Roots and Coefficients of Polynomials

You already know that for a quadratic \(ax^2 + bx + c = 0\), the sum of the roots is \(-\frac{b}{a}\) and the product is \(\frac{c}{a}\). In Further Maths, we extend this to cubics and quartics.

Relationships for a Cubic Equation: \(ax^3 + bx^2 + cx + d = 0\)

If the roots are \(\alpha\), \(\beta\), and \(\gamma\):

• Sum of roots: \(\sum \alpha = \alpha + \beta + \gamma = -\frac{b}{a}\)
• Sum of roots taken two at a time: \(\sum \alpha\beta = \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
• Product of roots: \(\alpha\beta\gamma = -\frac{d}{a}\)

Relationships for a Quartic Equation: \(ax^4 + bx^3 + cx^2 + dx + e = 0\)

If the roots are \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\):

• \(\sum \alpha = -\frac{b}{a}\)
• \(\sum \alpha\beta = \frac{c}{a}\)
• \(\sum \alpha\beta\gamma = -\frac{d}{a}\)
• \(\alpha\beta\gamma\delta = \frac{e}{a}\)

Memory Tip: Notice the signs always alternate! They start with negative \(b/a\), then positive \(c/a\), then negative \(d/a\), and so on. Think: "Minus, Plus, Minus, Plus."

Linear Transformations of Roots

Sometimes you’ll be asked to find a new equation where the roots are, for example, \(2\alpha, 2\beta, 2\gamma\).
Step-by-step process:
1. Let the new root be \(w = 2x\).
2. Rearrange for \(x\): \(x = \frac{w}{2}\).
3. Substitute this expression for \(x\) back into your original equation.
4. Simplify to get your new polynomial!

Quick Review: Polynomial roots and coefficients are linked by simple ratios. Always check the sign of your ratio based on the "alternating signs" rule!

2. Summation of Series

In standard Maths, you sum Arithmetic and Geometric series. Here, we look at sums of integers, squares, and cubes.

Standard Formulae

You need to know (and be able to use) these three sums:

• Sum of first \(n\) integers: \(\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)\)
• Sum of first \(n\) squares: \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)\)
• Sum of first \(n\) cubes: \(\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2\)

The Method of Differences

This is a clever trick to sum a series that doesn't fit a standard formula. If you can write a term \(u_r\) as the difference between two similar terms, like \(f(r) - f(r+1)\), the middle terms will "cancel out."

Imagine a row of falling dominoes. When the first one hits the second, and the second hits the third, only the very first push and the very last fall really matter. This is called a telescoping sum.

Common Mistake: When using the method of differences, be careful with the very first and very last terms. Don't forget to evaluate them properly!

3. Maclaurin Series

A Maclaurin Series is a way of writing complicated functions (like \(\sin x\) or \(e^x\)) as an infinite sum of simple powers of \(x\). This makes them much easier to work with in complex calculations.

Series you must recognise:

• \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...\) (Valid for all \(x\))
• \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...\) (Valid for all \(x\))
• \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...\) (Valid for all \(x\))
• \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...\) (Valid for \(-1 < x \le 1\))
• \((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ...\) (Valid for \(|x| < 1\))

Did you know? Computers actually use these series to calculate values for \(\sin(x)\) and \(\ln(x)\) because computers are very good at adding and multiplying, but they don't "know" trigonometry naturally!

Key Takeaway: Maclaurin series turn "curvy" functions into polynomials. Always check the range of validity (the values of \(x\) for which the series actually works).

4. Inequalities

Solving cubic and quartic inequalities is very similar to quadratics, but with more "regions" to consider.

Polynomial Inequalities

To solve something like \(x^3 - 4x > 0\):
1. Find the critical values by solving the equation as if it were an equals sign (\(x^3 - 4x = 0\)).
2. Sketch the graph. A positive cubic starts from the bottom left and goes to the top right.
3. Identify the regions on the \(x\)-axis where the graph is above zero.

Rational Inequalities

When solving \(\frac{ax+b}{cx+d} < ex+f\), NEVER just multiply by the denominator \((cx+d)\) because you don't know if it's positive or negative! If it's negative, the inequality sign must flip.

The Safe Way: Multiply both sides by the square of the denominator, \((cx+d)^2\). Since a square is always positive (or zero), the inequality sign stays the same. Then, move everything to one side and factorise.

5. Graphs of Rational Functions

Rational functions are fractions where both the top and bottom are polynomials. They often have asymptotes (lines the graph gets closer and closer to but never touches).

Finding Asymptotes for \(y = \frac{P(x)}{Q(x)}\)

• Vertical Asymptotes: These happen where the denominator \(Q(x) = 0\). (Because you can't divide by zero!)
• Horizontal Asymptotes: Look at what happens to \(y\) as \(x\) gets very large (\(x \to \infty\)). If the degrees of the top and bottom are the same, \(y\) will approach the ratio of the leading coefficients.

Using Quadratic Theory (D14)

For functions like \(y = \frac{ax^2+bx+c}{dx^2+ex+f}\), you can find the range of possible \(y\) values without using calculus.
1. Rearrange the equation into a quadratic in terms of \(x\). It will look like: \((...)x^2 + (...)x + (...) = 0\), where the coefficients contain \(y\).
2. Since \(x\) is a real number, the discriminant must be \(\ge 0\) (\(b^2 - 4ac \ge 0\)).
3. Solving this inequality for \(y\) tells you the maximum and minimum values of the function (the stationary points)!

6. Conic Sections and Transformations

Conic sections are a special family of curves. You need to be able to sketch these specific forms:

• Parabola: \(y^2 = 4ax\) (A curve opening to the right).
• Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (A stretched circle).
• Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) (Two separate "bowls" opening left and right).
• Rectangular Hyperbola: \(xy = c^2\) (The classic \(1/x\) shape but scaled).

Transformations: Remember your A-level transformation rules—they apply here too!
- \(f(x+k)\) is a translation left by \(k\).
- \(af(x)\) is a vertical stretch by scale factor \(a\).
- Replacing \(x\) with \(-x\) is a reflection in the \(y\)-axis.

Quick Summary: Conics are just specific shapes defined by their equations. Focus on finding where they cross the axes (intercepts) and where their asymptotes are (for hyperbolas).

Don't worry if this seems like a lot of information! The more you practice sketching these graphs and using the root formulae, the more natural it will feel. You've got this!