Further algebra and functions

Welcome to Further Algebra and Functions!

In this chapter, we take the algebra you learned in A Level Maths and turbocharge it. We’ll explore how the roots of equations relate to their coefficients, how to sum complex series using clever tricks, and how to sketch intricate graphs called Conics. While some of these topics might look intimidating at first, they are all built on patterns. Once you see the pattern, the math becomes much easier!

1. Roots 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 up to Quartic (degree 4) equations.

Key Relationships

For a cubic equation \(ax^3 + bx^2 + cx + d = 0\) with roots \(\alpha, \beta, \gamma\):

• Sum of roots: \(\sum \alpha = \alpha + \beta + \gamma = -\frac{b}{a}\)
• Sum of roots in pairs: \(\sum \alpha\beta = \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
• Product of roots: \(\alpha\beta\gamma = -\frac{d}{a}\)

Memory Aid: Notice the signs always alternate: Minus, Plus, Minus, Plus... and the denominator is always the leading coefficient \(a\).

Linear Transformations of Roots

Sometimes a question asks for a new equation where the roots are, for example, \(2\alpha+1, 2\beta+1, 2\gamma+1\).
Step-by-step:
1. Let \(w = 2x + 1\).
2. Rearrange to find \(x\): \(x = \frac{w-1}{2}\).
3. Substitute this expression for \(x\) back into your original equation.
4. Simplify to get your new equation in terms of \(w\).

Quick Review: The relationships between roots and coefficients allow us to find the properties of roots without actually solving the equation!

2. Summation of Series

You will learn specific formulas to sum integers, squares, and cubes. These are your "building blocks."

Standard Formulae

• \(\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)\)
• \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)\)
• \(\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2\) (Notice this is just the square of the first formula!)

The Method of Differences

Don't worry if a series looks impossible to sum! If you can write the general term in the form \(f(r) - f(r+1)\), most terms will cancel out. This is often called a Telescoping Series.

Example: Think of a line of people where each person gives £5 to the person in front of them. Only the first and last person actually end up with a change in their wallet because everyone in the middle receives and gives away the same amount!

Common Mistake: When using the method of differences, be careful with the limits. Always write out the first few terms and the last few terms to see exactly what cancels.

Key Takeaway: Complex sums can be broken down into standard formulas or simplified through cancellation (method of differences).

3. Maclaurin Series

A Maclaurin series is a way to represent complicated functions (like \(\sin x\) or \(e^x\)) as an infinite sum of polynomials. This makes them much easier to work with in calculators and computers.

The General Form

\(f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \dots + \frac{x^r}{r!}f^{(r)}(0) + \dots\)

Standard Series to Memorise

• \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots\)
• \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\)
• \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\)

Did you know? These approximations are only valid for certain values of \(x\) (the range of validity). For example, \(\ln(1+x)\) only works when \(-1 < x \le 1\).

Evaluation of Limits: You can use these series or l'Hôpital's rule to find limits that look like \(\frac{0}{0}\). L'Hôpital's rule says: if \(\frac{f(x)}{g(x)}\) gives \(\frac{0}{0}\), differentiate the top and bottom separately and try again!

4. Inequalities

Solving Further Maths inequalities requires more care than standard GCSE algebra. Never multiply both sides by an expression involving \(x\) (like \(x-2\)) because you don't know if it's positive or negative!

The "Square the Denominator" Trick

To solve \(\frac{ax+b}{cx+d} < ex+f\):
1. Multiply both sides by \((cx+d)^2\). Since a square is always positive, the inequality sign stays the same.
2. Move everything to one side to get a cubic or quartic inequality.
3. Find the critical values (roots).
4. Sketch the graph or use a number line to find the correct regions.

Quick Review: Always sketch a graph for inequalities. It’s the safest way to ensure you pick the right intervals (e.g., between two points or outside of them).

5. Rational Functions and Graphs

Rational functions are fractions where the top and bottom are polynomials, such as \(y = \frac{ax+b}{cx+d}\).

Asymptotes

Asymptotes are lines that the graph approaches but never touches (usually). Think of them as "invisible walls."

• Vertical Asymptotes: Occur where the denominator is zero.
• Horizontal Asymptotes: Look at what happens as \(x\) becomes very large.
• Oblique (Slanted) Asymptotes: Occur when the degree of the numerator is exactly one higher than the denominator. Use long division to find the equation of the line.

Using the Discriminant for Range

If you need to find the possible \(y\)-values (the range) of a rational function without using calculus:
1. Set \(y = \frac{f(x)}{g(x)}\).
2. Rearrange into a quadratic equation in \(x\): \(Ax^2 + Bx + C = 0\).
3. Use the condition \(b^2 - 4ac \ge 0\) (because \(x\) must be a real number).
4. Solve this new inequality to find the range of \(y\).

6. Conic Sections

Conics are shapes made by slicing a cone. You need to know their standard equations and how to sketch them.

Types of Conics

• Parabola: \(y^2 = 4ax\). It looks like a "cup" on its side.
• 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" facing away from each other.
• Rectangular Hyperbola: \(xy = c^2\). The standard \(1/x\) shape you know from GCSE.

Transformations

You can move these shapes around just like any other function:

• \(f(x-a)\): Shift right by \(a\).
• \(f(x)+a\): Shift up by \(a\).
• \(af(x)\): Vertical stretch by scale factor \(a\).

Key Takeaway: Conics are just specific algebraic patterns. Identifying the "standard form" is 90% of the work!

Final Summary

1. Roots: Use \(\sum \alpha = -b/a\) and alternating signs.
2. Series: Use standard formulas or the method of differences to cancel terms.
3. Maclaurin: Use the formula to turn functions into polynomials.
4. Inequalities: Multiply by the square of the denominator to avoid sign errors.
5. Graphs: Find asymptotes first, and use the discriminant to find stationary points.
6. Conics: Learn the four standard equations and how to apply translations and stretches.

Don't worry if this seems tricky at first—algebra is a skill that gets much better with practice. Keep sketching those graphs and the patterns will start to feel like second nature!