Roots and Coefficients Relationship

Introduction and Key Concepts

A solid grasp of polynomials is essential for solving real-world problems effectively. Here's why:

• Polynomials: Composed of variables and coefficients.
• Roots and Coefficients: Understanding these helps predict and apply polynomial behaviour in real-world scenarios.
• Importance: Mastering these concepts is vital for efficient modelling and solving of challenges such as cost optimisation and environmental assessments.

1. Quadratic Equations

Introduction to Quadratic Equations

Quadratic Equation: in standard form: $ax^2 + bx + c = 0$.

Understanding Vieta's Formulas

Vieta's Formulas for Quadratics:

• Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
• Product of the roots: $\alpha\beta = \frac{c}{a}$

Common Mistakes

• Sign Errors: Incorrect interpretation of signs in Vieta's formulas.
• Forgetting Normalisation: Misapplying $a \neq 1$ as if $a=1$.

Example Problems

Problem 1:

Solve $2x^2 - 3x + 1 = 0$ using Vieta's formulas:

1. Identify coefficients: $a=2$, $b=-3$, $c=1$.
2. Apply Vieta's formulas:
   • $\alpha + \beta = \frac{3}{2}$
   • $\alpha\beta = \frac{1}{2}$
3. Using these relationships to find the roots:
   • Let's use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
   • $x = \frac{3 \pm \sqrt{9-8}}{4} = \frac{3 \pm \sqrt{1}}{4} = \frac{3 \pm 1}{4}$
   • Therefore, $x = 1$ or $x = \frac{1}{2}$
4. Verification: $1 + \frac{1}{2} = \frac{3}{2}$ and $1 \times \frac{1}{2} = \frac{1}{2}$

Problem 2:

Construct $x^2 - 5x + 6 = 0$ using roots $\alpha = 2$ and $\beta = 3$.

1. Set $(x - \alpha)(x - \beta) = 0$
2. Expand: $(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6 = 0$

Exercises with Hints and Solutions

• Exercise 1: Determine the sums and products of the roots for $x^2 + 5x + 6 = 0$.

  • Solution: With $a=1$, $b=5$, $c=6$:
    • Sum of roots = $-\frac{b}{a} = -5$
    • Product of roots = $\frac{c}{a} = 6$

• Exercise 2: Formulate a quadratic equation with $\alpha + \beta = -7$ and $\alpha \beta = 10$.

  • Solution:
    • Since $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$
    • With $a=1$: $-\frac{b}{1} = -7$, thus $b = 7$
    • And $\frac{c}{1} = 10$, thus $c = 10$
    • The equation is $x^2 + 7x + 10 = 0$

2. Cubic Equations

Introduction to Cubic Equations

A cubic equation is a third-degree polynomial, expressed as:

$ax^3 + bx^2 + cx + d = 0$

• Role of Coefficients: Each coefficient significantly influences the graph of the polynomial.
• Roots: Solutions are represented as $\alpha, \beta, \gamma$.

Vieta's Formulas in Cubics

Vieta's Formulas:

• Sum of the Roots: $\alpha + \beta + \gamma = -\frac{b}{a}$
• Sum of Products: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$
• Total Product: $\alpha\beta\gamma = -\frac{d}{a}$

Common Mistakes and Tips

• Sign errors and failure to account for all coefficients.

Worked Examples

Example 1 - Derivation from Known Roots

• Given roots: $\alpha = 1$, $\beta = 2$, $\gamma = 3$.
• Use Vieta's to calculate:
  • Sum of roots: $\alpha + \beta + \gamma = 1 + 2 + 3 = 6$
  • Sum of products: $\alpha\beta + \beta\gamma + \gamma\alpha = 1 \times 2 + 2 \times 3 + 3 \times 1 = 2 + 6 + 3 = 11$
  • Product of roots: $\alpha\beta\gamma = 1 \times 2 \times 3 = 6$
• Therefore, the cubic equation is: $x^3 - 6x^2 + 11x - 6 = 0$

General Polynomial Form

• General Form: $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0$.
• Roots: Solutions $r_1, r_2, \ldots, r_n$ where the polynomial equals zero.

• Vieta's Formulas for a polynomial of degree n:
  • Sum of Roots: $r_1 + r_2 + \ldots + r_n = -\frac{a_{n-1}}{a_n}$
  • Product of Roots: $r_1 r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n}$

Constructing Polynomials from Roots

Objective: Deriving Polynomials with Known Roots

Example:

• Given Roots $x_1 = 2 + i$, $x_2 = 2 - i$.
• Process:
  • Identify conjugates.
  • Form the polynomial: $(x-(2+i))(x-(2-i))$
  • Expand: $(x-2-i)(x-2+i)$
  • Further expansion: $(x-2)^2 - i^2 = (x-2)^2 + 1$
  • Final form: $x^2 - 4x + 4 + 1 = x^2 - 4x + 5$

Problem-Solving Exercises

• Economic Modelling: Analyse $x^3 - 4x^2 - x + 6$.

  • Solution:
    • Using Vieta's formulas for cubic equations:
    • Sum of roots = $\frac{4}{1} = 4$
    • Sum of products = $\frac{-(-1)}{1} = 1$
    • Product of roots = $\frac{-6}{1} = -6$
    • The roots must satisfy these relationships.

• Physics Applications: Examine projectile trajectories.

  • Solution:
    • If a projectile's path is modelled by $y = ax^2 + bx + c$,
    • The points where it touches the ground have $y = 0$
    • Those x-coordinates are roots of $ax^2 + bx + c = 0$
    • Using Vieta's, sum of roots = $-\frac{b}{a}$, product of roots = $\frac{c}{a}$
    • This can help determine landing positions based on launch parameters.

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Exam Tips

• Regular Practice: Solving practice problems enhances comprehension.
• Consistent Verification: Ensure calculations comply with Vieta's formulas and are verified.
• Visual Aids: Utilise diagrams to conceptualise and validate solutions.