Welcome to Core Pure Algebra!
In this chapter, we are going to explore the secret "backdoor" connections between the roots of an equation (the answers you get when you solve it) and its coefficients (the numbers in front of the \(x\) terms). Instead of solving a long cubic equation to find its roots, you’ll learn how to find out a lot about those roots just by looking at the equation itself!
This is a fundamental skill in Further Mathematics because it allows us to build new equations and solve complex problems without doing the "heavy lifting" of long division or factorisation every time.
1. Roots and Coefficients: The DNA of an Equation
Think of the coefficients of a polynomial as the "DNA" and the roots as the "physical traits." Just as DNA determines how someone looks, these coefficients determine exactly where the roots are. For the Further Maths syllabus, we focus on Quadratic, Cubic, and Quartic equations.
The General Polynomial
We usually write our equations in this form:
\(ax^n + bx^{n-1} + cx^{n-2} + ... = 0\)
The Quadratic Case (Degree 2)
For \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\):
1. Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
2. Product of roots: \(\alpha\beta = \frac{c}{a}\)
The Cubic Case (Degree 3)
For \(ax^3 + bx^2 + cx + d = 0\) with roots \(\alpha\), \(\beta\), and \(\gamma\):
1. Sum of roots: \(\alpha + \beta + \gamma = -\frac{b}{a}\)
2. Sum of roots taken two at a time: \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
3. Product of roots: \(\alpha\beta\gamma = -\frac{d}{a}\)
The Quartic Case (Degree 4)
For \(ax^4 + bx^3 + cx^2 + dx + e = 0\) with roots \(\alpha, \beta, \gamma, \delta\):
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: \(\alpha\beta\gamma\delta = \frac{e}{a}\)
Memory Aid: The Sign Seesaw
Don’t worry about memorising every single formula individually! Just remember the Alternating Sign Rule. The denominator is always \(a\). The numerators follow the coefficients (\(b, c, d, e\)) but the signs always alternate, starting with a minus:
Minus (\(-\frac{b}{a}\)), Plus (\(\frac{c}{a}\)), Minus (\(-\frac{d}{a}\)), Plus (\(\frac{e}{a}\)).
Quick Review:
The symbol \(\sum \alpha\beta\) is just a lazy (and efficient!) way of saying "multiply every possible pair of roots together and add them up." For a cubic, that means \(\alpha\beta + \beta\gamma + \gamma\alpha\).
2. Transforming the Roots
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 3 bigger than the original ones. The syllabus focuses on linear transformations.
Example: Shifting Roots
Suppose the roots of \(x^3 - 2x^2 + 5x - 1 = 0\) are \(\alpha, \beta, \gamma\). Find an equation with roots \((\alpha+2), (\beta+2), (\gamma+2)\).
Step-by-Step Process:
1. Let our new root be \(w\). So, \(w = x + 2\).
2. Rearrange this to find \(x\) in terms of \(w\): \(x = w - 2\).
3. Substitute this back into the original equation wherever you see an \(x\):
\((w-2)^3 - 2(w-2)^2 + 5(w-2) - 1 = 0\)
4. Expand and simplify! (This gives you the new equation in terms of \(w\)).
Example: Scaling Roots
If you want roots that are double the original (\(w = 2x\)), you would substitute \(x = \frac{w}{2}\) into your equation.
Common Mistake to Avoid:
The "Sign Flip" Trap: When substituting \(w = x + 3\), many students accidentally substitute \(x = w + 3\). Always rearrange the transformation equation for \(x\) first before plugging it in!
3. Summary and Key Takeaways
Key Points:
• For any polynomial, the sum of roots is always \(-\frac{b}{a}\).
• The product of roots is the last coefficient divided by \(a\), but you must check the sign! (Positive if even degree, negative if odd degree).
• To create a new equation with transformed roots, use the substitution method: define \(w\), rearrange for \(x\), and plug it back in.
Did you know?
These relationships are known as Vieta's Formulas, named after the 16th-century French mathematician François Viète. He was one of the first people to use letters to represent numbers in equations, which is why your algebra homework looks the way it does today!
Don't worry if the expansions for quartics feel a bit long at first—once you master the substitution method, the logic is always the same!