Theory of Equations

Theory of Equations: The Secret Language of Roots 🌱

Imagine you’re a detective. Every polynomial equation is a mystery box, and the roots are the hidden treasures inside. Today, we’ll learn the secret codes that tell us about these treasures—even before we find them!

The Big Picture: What Are We Learning?

Think of a polynomial equation like a recipe. The roots are the special ingredients that make the recipe work (make the equation equal zero). Instead of finding each root one by one (which is hard!), mathematicians discovered clever shortcuts. These shortcuts let us know things about the roots without solving the equation!

graph TD
    A["Polynomial Equation"] --> B["Sum of Roots"]
    A --> C["Product of Roots"]
    A --> D["Symmetric Functions"]
    A --> E["Building Equations"]
    A --> F["Common Roots"]
    A --> G["Root Location"]
    A --> H["Sign Changes"]
    A --> I["Transformations"]

1. Sum of Roots: Adding Up the Treasures

The Story

Imagine you have a box with 3 gold coins inside. You can’t open the box, but there’s a label that tells you the total weight of all coins. That’s what sum of roots does—it tells you what all roots add up to!

The Magic Formula

For a quadratic equation: ax² + bx + c = 0

Sum of roots = −b/a

For a cubic equation: ax³ + bx² + cx + d = 0

Sum of roots = −b/a

Simple Example

Equation: x² − 5x + 6 = 0

Here: a = 1, b = −5, c = 6

Sum of roots = −(−5)/1 = 5

Let’s check! The roots are x = 2 and x = 3. Sum = 2 + 3 = 5 ✓

The Pattern

The sum of roots always equals −(second coefficient)/(first coefficient)

2. Product of Roots: Multiplying the Treasures

The Story

Now imagine the label on your treasure box also tells you what you’d get if you multiplied all the coins’ values together. That’s the product of roots!

The Magic Formula

For a quadratic equation: ax² + bx + c = 0

Product of roots = c/a

For a cubic equation: ax³ + bx² + cx + d = 0

Product of roots = −d/a (notice the sign changes!)

Simple Example

Equation: x² − 5x + 6 = 0

Here: a = 1, c = 6

Product of roots = 6/1 = 6

Let’s check! The roots are x = 2 and x = 3. Product = 2 × 3 = 6 ✓

Quick Reference Table

3. Symmetric Root Functions: Fair Play

The Story

Imagine you have twins who always share equally. If one gets a gift, the other gets the same. Symmetric functions treat all roots fairly—swap any two roots, and the answer stays the same!

What Are They?

If α and β are roots:

• α + β is symmetric (swap them: β + α = same!)
• α × β is symmetric (swap them: β × α = same!)
• α² + β² is symmetric
• α³ + β³ is symmetric

The Clever Trick

You can find α² + β² without knowing α and β individually!

Formula: α² + β² = (α + β)² − 2αβ

Example

For x² − 5x + 6 = 0:

• α + β = 5
• αβ = 6

α² + β² = (5)² − 2(6) = 25 − 12 = 13

Check: 2² + 3² = 4 + 9 = 13 ✓

More Symmetric Functions

4. Equations from Roots: Building Backwards

The Story

You’re now the chef, not the detective! If someone gives you the roots, can you build the equation? Absolutely!

The Magic Recipe

If your roots are α and β, your equation is:

x² − (sum of roots)x + (product of roots) = 0

Or simply: x² − (α + β)x + αβ = 0

Example: Build an Equation

Given roots: 4 and −2

• Sum = 4 + (−2) = 2
• Product = 4 × (−2) = −8

Equation: x² − 2x − 8 = 0

For Three Roots

If roots are α, β, γ:

x³ − (α+β+γ)x² + (αβ+βγ+γα)x − αβγ = 0

Example: Three Roots

Given roots: 1, 2, 3

• Sum = 1 + 2 + 3 = 6
• Sum of products in pairs = (1×2) + (2×3) + (3×1) = 2 + 6 + 3 = 11
• Product of all = 1 × 2 × 3 = 6

Equation: x³ − 6x² + 11x − 6 = 0

5. Common Roots: Shared Secrets

The Story

Two treasure boxes might share the same coin inside! When two equations share a root, we call it a common root.

How to Find Common Roots

Method 1: Substitution If α is a common root of two equations, it satisfies both. Substitute and solve!

Method 2: Resultant Set up a system and eliminate to find the common value.

Example

Find the common root of:

• x² − 5x + 6 = 0 (roots: 2, 3)
• x² − 4x + 4 = 0 (roots: 2, 2)

The common root is x = 2!

The Condition for Common Root

For equations:

• a₁x² + b₁x + c₁ = 0
• a₂x² + b₂x + c₂ = 0

If they share one common root α:

(c₁a₂ − c₂a₁)² = (b₁c₂ − b₂c₁)(a₁b₂ − a₂b₁)

6. Location of Roots: Where Are They Hiding?

The Story

Sometimes we don’t need the exact roots—we just need to know where they are. Are they positive? Negative? Between 1 and 5? This is like knowing “the treasure is buried somewhere in the garden” without knowing the exact spot!

Key Principles

For f(x) = ax² + bx + c with roots α and β:

Both roots positive (α > 0, β > 0) when:

• Sum (−b/a) > 0
• Product (c/a) > 0
• Discriminant (b² − 4ac) ≥ 0

Both roots negative (α < 0, β < 0) when:

• Sum (−b/a) < 0
• Product (c/a) > 0
• Discriminant ≥ 0

Roots of opposite signs when:

• Product (c/a) < 0

The Interval Test

To check if roots lie between numbers p and q:

• f(p) and f(q) should have opposite signs (one positive, one negative)

Example

Does x² − 4x + 3 = 0 have roots between 0 and 5?

• f(0) = 0 − 0 + 3 = 3 (positive)
• f(5) = 25 − 20 + 3 = 8 (positive)

Wait, both positive! But let’s check f(1) and f(4):

• f(1) = 1 − 4 + 3 = 0 (root at 1!)
• f(3) = 9 − 12 + 3 = 0 (root at 3!)

Yes, both roots (1 and 3) are between 0 and 5!

7. Descartes’ Rule of Signs: Counting Without Solving

The Story

René Descartes discovered a magical shortcut. By just counting how many times the signs (+/−) change in an equation, you can predict the maximum number of positive or negative roots!

The Rule

For positive roots: Count sign changes in f(x) For negative roots: Count sign changes in f(−x)

Step-by-Step Example

Equation: f(x) = x³ − 2x² − x + 2

Signs: + − − +

Changes: + to − (1), − to − (no change), − to + (2)

Sign changes = 2

So there are at most 2 positive roots (could be 2 or 0).

Now check f(−x): (−x)³ − 2(−x)² − (−x) + 2 = −x³ − 2x² + x + 2

Signs: − − + +

Changes: − to − (no), − to + (1), + to + (no)

Sign changes = 1

So there is exactly 1 negative root.

Key Insight

The actual number of positive roots equals the number of sign changes, OR that number minus an even number.

8. Transformation of Equations: Shape-Shifting

The Story

What if you could take an equation and transform it? Like turning a caterpillar into a butterfly! You can create new equations whose roots are related to the original roots in special ways.

Common Transformations

1. Roots increased by k: Replace x with (x − k)

Original roots: α, β New roots: α + k, β + k

2. Roots multiplied by k: Replace x with x/k

Original roots: α, β New roots: kα, kβ

3. Reciprocal roots: Replace x with 1/x, then multiply through

Original roots: α, β New roots: 1/α, 1/β

4. Square roots: Replace x with x²

Original roots: α, β New roots: α², β²

Example: Increase Roots by 2

Original: x² − 5x + 6 = 0 (roots: 2, 3)

Replace x with (x − 2): (x − 2)² − 5(x − 2) + 6 = 0 x² − 4x + 4 − 5x + 10 + 6 = 0 x² − 9x + 20 = 0

New roots: 4 and 5 (which are 2+2 and 3+2!) ✓

Example: Reciprocal Roots

Original: 2x² − 5x + 3 = 0

Replace x with 1/x: 2(1/x)² − 5(1/x) + 3 = 0 2/x² − 5/x + 3 = 0

Multiply by x²: 3x² − 5x + 2 = 0

Notice how coefficients reversed (almost)!

Summary: Your Detective Toolkit 🔍

graph TD
    A["Theory of Equations"] --> B["Know About Roots"]
    A --> C["Build Equations"]
    A --> D["Transform Equations"]

    B --> B1["Sum = -b/a"]
    B --> B2["Product = c/a or ±d/a"]
    B --> B3["Symmetric Functions"]
    B --> B4["Common Roots"]
    B --> B5["Location Tests"]
    B --> B6["Sign Counting"]

    C --> C1["Use Sum & Product"]

    D --> D1["Shift: x → x-k"]
    D --> D2["Scale: x → x/k"]
    D --> D3["Reciprocal: x → 1/x"]

The Golden Rules to Remember

1. Sum of roots = −(second coefficient)/(first coefficient)
2. Product of roots = (last coefficient)/(first coefficient) × (±1 based on degree)
3. Symmetric functions can be computed from sum and product
4. Building equations = x² − (sum)x + (product) = 0
5. Common roots satisfy both equations simultaneously
6. Location is tested using signs of f(p) and f(q)
7. Descartes’ Rule = count sign changes for max positive/negative roots
8. Transform by replacing x with (x−k), x/k, 1/x, or x²

You’re now equipped to understand polynomial equations like never before. The roots might be hidden, but their secrets are no longer mysterious to you! 🎓