Welcome to Further Algebra!
In your standard A Level Maths, you spend a lot of time finding the roots of equations (solving for \(x\)). In Further Mathematics, we take a shortcut! Instead of solving the whole equation, we look at the secret relationships between the roots (the answers) and the coefficients (the numbers in front of the \(x\) terms).
This chapter is part of the Pure Core section of your OCR AS Level course. It is incredibly useful because it allows us to build new equations or find out properties of unknown roots without doing the "heavy lifting" of full algebraic long division or complex factoring. Don't worry if this seems a bit abstract at first—once you see the pattern, it becomes much like following a recipe!
1. Roots and Coefficients
Every polynomial equation has a relationship between the numbers used to build it (the coefficients) and its solutions (the roots). We usually name the roots using Greek letters like \(\alpha\) (alpha), \(\beta\) (beta), \(\gamma\) (gamma), and \(\delta\) (delta).
The Basics: Quadratics (A Quick Reminder)
You might remember this from GCSE or standard Maths. For a quadratic equation \(ax^2 + bx + c = 0\):
1. The sum of the roots: \(\alpha + \beta = -\frac{b}{a}\)
2. The product of the roots: \(\alpha\beta = \frac{c}{a}\)
Stepping Up: Cubic Equations
For a cubic equation \(ax^3 + bx^2 + cx + d = 0\), we have three roots: \(\alpha\), \(\beta\), and \(\gamma\).
The relationships are:
1. Sum of roots: \(\sum \alpha = \alpha + \beta + \gamma = -\frac{b}{a}\)
2. Sum of roots in pairs: \(\sum \alpha\beta = \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
3. Product of roots: \(\alpha\beta\gamma = -\frac{d}{a}\)
The Big One: Quartic Equations
For a quartic equation \(ax^4 + bx^3 + cx^2 + dx + e = 0\), we have four roots: \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\).
1. Sum of roots: \(\sum \alpha = -\frac{b}{a}\)
2. Sum of roots in pairs: \(\sum \alpha\beta = \frac{c}{a}\)
3. Sum of roots in triples: \(\sum \alpha\beta\gamma = -\frac{d}{a}\)
4. Product of all roots: \(\alpha\beta\gamma\delta = \frac{e}{a}\)
Did you know? These are known as Vieta's Formulas, named after the 16th-century French mathematician François Viète.
Memory Aid: The Sign Flip Trick
Notice the pattern for the signs of the results: it always starts with negative, then positive, then negative, then positive.
\(-\frac{b}{a}\), \(+\frac{c}{a}\), \(-\frac{d}{a}\), \(+\frac{e}{a}\)...
Just remember: Start with minus, then alternate!
Quick Review: Important Points
- The first term (\(a\)) is always the denominator (on the bottom).
- The sum of roots is always related to the second term (\(b\)).
- Always ensure your equation is set to equal zero before identifying coefficients.
Common Mistake: Forgetting the minus sign on the first relationship! Always double-check if your \(b\) value is already negative, as "negative of a negative" makes a positive.
Key Takeaway: You can find the sum and product of roots for any polynomial up to a quartic just by looking at the coefficients \(a, b, c, d, e\).
2. Transforming Equations
Sometimes, we are given an equation and asked to find a new equation where the roots are slightly different—for example, where every root is doubled, or every root is increased by 1.
Think of this like moving a graph. If you know where the "zeros" are on one graph, and you shift the whole graph to the right, you have created a new equation with new "zeros."
Step-by-Step: The Substitution Method
Suppose you have an equation in terms of \(x\) with roots \(\alpha, \beta, \gamma\). You want a new equation in terms of \(w\) where the roots are \(\alpha + 2, \beta + 2, \gamma + 2\).
Step 1: Define the relationship between the old roots (\(x\)) and the new roots (\(w\)).
In this case: \(w = x + 2\)
Step 2: Rearrange this to make \(x\) the subject.
\(x = w - 2\)
Step 3: Substitute this expression for \(x\) into your original equation.
If the original was \(x^3 + 3x^2 - 4 = 0\), the new one is \((w - 2)^3 + 3(w - 2)^2 - 4 = 0\).
Step 4: Expand and simplify to get your new equation in terms of \(w\).
Common Transformations to Know
- Roots are \(k\) times the original: Let \(w = kx\), so substitute \(x = \frac{w}{k}\).
- Roots are squares of the original: Let \(w = x^2\), so substitute \(x = \sqrt{w}\) (be careful with the algebra here!).
- Roots are the reciprocal: Let \(w = \frac{1}{x}\), so substitute \(x = \frac{1}{w}\).
Analogy: Imagine you are a tailor. If you know a suit fits one person, and you need to make a suit for someone 2 inches taller, you just add 2 inches to all the original measurements. You don't need to start from scratch!
Quick Review: Common Mistakes to Avoid
- Getting the substitution backwards: If the new roots are \(\alpha + 5\), you must substitute \(x = w - 5\), not \(w + 5\).
- Algebra errors: Squaring or cubing brackets like \((w - 2)^3\) is where most marks are lost. Take your time!
- Missing terms: If an equation like \(x^3 + 2x - 1 = 0\) is missing an \(x^2\) term, remember that the coefficient \(b\) is actually zero.
Key Takeaway: Use the substitution \(w = [new \ root]\) to find a brand-new equation without ever needing to find the actual values of \(\alpha\) or \(\beta\).
Summary Checklist
Before you move on to practice questions, make sure you can:
- State the Sum (\(\sum \alpha\)), Pair-Sum (\(\sum \alpha\beta\)), and Product (\(\alpha\beta\gamma...\)) for quadratics, cubics, and quartics.
- Correctly apply the alternating signs (\(-, +, -, +\)).
- Use the substitution method to transform an equation's roots.
- Simplify complex algebraic brackets accurately.
Don't worry if the algebra feels long! With practice, the patterns will become second nature. You've got this!