Further algebra and functions

Welcome to Further Algebra and Functions!

In this chapter, we are going to dive deeper into the behavior of polynomial equations and explore how we can represent complex functions as simple, infinite sums. Think of this as looking "under the hood" of the functions you've used since GCSE. We'll learn how roots of equations relate to their coefficients, how to sum up long series of numbers quickly, and how to turn tricky functions like \(\sin(x)\) into easy-to-use polynomials.

Don't worry if some of these symbols look intimidating at first. We’ll break them down step-by-step. Let’s get started!

1. Roots and Coefficients of Polynomials

Usually, we find the roots (the solutions) of an equation. Here, we do the opposite: we look at the coefficients (the numbers in front of the variables) to figure out things about the roots without actually solving the equation!

The Basics: Quadratic, Cubic, and Quartic

For any polynomial in the form \(ax^n + bx^{n-1} + cx^{n-2} + \dots = 0\), there are patterns connecting the roots (\(\alpha, \beta, \gamma, \delta\)) and the coefficients (\(a, b, c, d, e\)).

The "Sum of Roots" Rule:
The sum of the roots is always \(-\frac{b}{a}\).
• For a Cubic (\(ax^3 + bx^2 + cx + d = 0\)): \(\alpha + \beta + \gamma = -\frac{b}{a}\)
• For a Quartic (\(ax^4 + bx^3 + cx^2 + dx + e = 0\)): \(\alpha + \beta + \gamma + \delta = -\frac{b}{a}\)

The "Pairs" Rule:
The sum of roots multiplied in pairs is always \(\frac{c}{a}\).
• For a Cubic: \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)

The Product Rule:
The product of all roots alternates sign based on the degree of the equation:
• Cubic (odd degree): \(\alpha\beta\gamma = -\frac{d}{a}\)
• Quartic (even degree): \(\alpha\beta\gamma\delta = \frac{e}{a}\)

Memory Aid: The signs always alternate! Start with \(-\frac{b}{a}\), then \(+\frac{c}{a}\), then \(-\frac{d}{a}\), and so on. It’s like a "Minus-Plus-Minus" rhythm.

Evaluating Expressions

Sometimes the exam will ask you to find the value of \(\alpha^2 + \beta^2 + \gamma^2\). You don't need the roots! You just need this identity:
\(\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)\)
Analogy: It’s like knowing the total weight of a cake and the weight of the frosting—you can figure out the weight of the sponge without seeing it!

Common Mistake to Avoid: When using the formulas, always make sure your equation is set to equal zero first. Also, double-check your signs; missing a negative sign is the most common way to lose marks here.

Key Takeaway:

The coefficients of a polynomial tell a story about its roots. Use the alternating sign rule (\(-\frac{b}{a}, \frac{c}{a}, -\frac{d}{a}\dots\)) to extract that story.

2. Linear Transformations of Roots

What if we have an equation with roots \(\alpha, \beta, \gamma\), and we want a new equation where the roots are all 3 bigger (e.g., \(\alpha+3, \beta+3, \gamma+3\))?

The Step-by-Step Process:
1. Let the new root be \(w\). So, \(w = x + 3\).
2. Rearrange this to find \(x\) in terms of \(w\): \(x = w - 3\).
3. Substitute this expression back into the original equation wherever you see an \(x\).
4. Expand and simplify to get your new equation in \(w\).

Example: If you are told the roots are \(2\alpha, 2\beta, 2\gamma\), you substitute \(x = \frac{w}{2}\) into the original equation.

Key Takeaway:

To change the roots, substitute the "inverse" operation into the equation. If you want to double the roots, replace \(x\) with \(\frac{x}{2}\).

3. Summation of Series

You already know how to sum arithmetic and geometric series. In Further Maths, we learn formulas for the sums of squares (\(r^2\)) and cubes (\(r^3\)).

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

Did you know? The sum of cubes is actually just the (sum of integers)\(^2\)! If you remember the first one, the third one is easy.

Trick for Tough Problems: When summing something like \(\sum r(r+2)\), split it into \(\sum r^2 + \sum 2r\). Then apply the formulas separately.
Top Tip: Always look for common factors like \(n\) and \((n+1)\) at the very start of your simplification. It saves you from having to expand massive brackets!

Key Takeaway:

Complex series can be broken down into simpler parts. Use the standard formulas and factorize as early as possible to keep the algebra clean.

4. The Method of Differences

This is a clever way to sum a series where almost everything cancels out. It is often called a "Telescoping Series."

Analogy: Imagine a collapsible camping telescope. When you push it together, all the middle parts disappear, and you’re just left with the two ends. That’s exactly how this works!

How to do it:
1. Use Partial Fractions to split the term into two parts (usually one subtracted from the other).
2. Write out the first few terms of the sum (let \(r=1, r=2, r=3\)).
3. Write out the last few terms (let \(r=n-1, r=n\)).
4. Cross out the terms that appear as both positive and negative.
5. Sum what is left over.

Example: If the general term is \(\frac{1}{r} - \frac{1}{r+1}\), the \(\frac{1}{2}\) from the first term will cancel with the \(-\frac{1}{2}\) from the second term. Eventually, you’ll just be left with the very first and very last bits.

Key Takeaway:

If you see a sum involving fractions, try Partial Fractions first. If terms start canceling out, you’re using the Method of Differences!

5. Maclaurin Series

A Maclaurin Series allows us to turn complicated functions like \(\ln(1+x)\) or \(e^x\) into an infinite polynomial. This is useful because polynomials are much easier to differentiate and integrate.

The General Formula

\(f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots + \frac{f^{(r)}(0)}{r!}x^r + \dots\)

Step-by-Step for Finding a Series:
1. Differentiate the function multiple times (\(f', f'', f'''\)).
2. Plug \(x=0\) into the function and all its derivatives.
3. Substitute these values into the formula above.
4. Simplify the coefficients.

Standard Series to Know

You should be familiar with these (they are in your formula book, but knowing them helps!):
• \(e^x = 1 + x + \frac{x^2}{2!} + \dots\)
• \(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\) (Only odd powers!)
• \(\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\) (Only even powers!)
• \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots\) (No factorials in the denominator! Valid for \(-1 < x \le 1\))

Quick Review Box:
• \(\sin(x)\) is an odd function \(\rightarrow\) odd powers.
• \(\cos(x)\) is an even function \(\rightarrow\) even powers.
• Always use radians for Maclaurin series involving trigonometry!

Compound Functions:
If you need the series for \(e^{2x}\), you don't need to redo the whole process. Just take the standard \(e^x\) series and replace every \(x\) with \(2x\). Easy!

Key Takeaway:

Maclaurin series are "approximations." The more terms you add, the more accurate the polynomial becomes to the original function. Use standard series as shortcuts whenever possible.

Summary of Further Algebra

• Roots: Use \(-\frac{b}{a}, \frac{c}{a}, \dots\) and substitute \(w\) for transformations.
• Series: Use standard formulas for \(\sum r^2\) and \(\sum r^3\), and look for the "telescoping" cancelation in the Method of Differences.
• Maclaurin: Use the value of derivatives at \(x=0\) to build polynomials for complex functions.

Don't worry if this seems tricky at first—algebra is a skill that gets much easier with practice. Keep working through the examples, and you'll soon start seeing the patterns!