Theory of Equations | SuccessSetu

🔹 Polynomial

A polynomial is an algebraic expression of the form:

a₀ + a₁x + a₂x² + ... + aₙxⁿ

Where n is a natural number.

🔹 Real Polynomial

If all coefficients and variable x are real, then it is called real polynomial.

Example: 3x² + 2x + 5

🔹 Degree of Polynomial

Highest power of x is called degree.

f(x)=5x⁴+2x²+1 → Degree = 4

🔹 Types of Polynomial

Linear → Degree 1

Quadratic → Degree 2

Cubic → Degree 3

Biquadratic → Degree 4

🔹 Polynomial Equation

If f(x)=0 then it is called polynomial equation.

Example: x²+3x+2=0

🔹 Quadratic Equation

General form:

ax² + bx + c = 0 (a ≠ 0)

🔹 Roots of Equation

Values of x which satisfy equation are called roots.

If x²-5x+6=0 → Roots = 2,3

🔹 Factorisation Method

ax²+bx+c = a(x-α)(x-β)

Then roots are α and β.

🔹 Direct Formula

x = (-b ± √(b²-4ac)) / 2a

Discriminant:

D = b² - 4ac

🔹 Nature of Roots

D > 0 → Two Real Roots

D = 0 → Equal Roots

D < 0 → Imaginary Roots

🔹 Sridharacharya Formula

Quadratic formula is also known as Sridharacharya Formula.

🔹 Given Quadratic Equation

ax² + bx + c = 0

General quadratic equation.

🔹 Condition for Common Root

(ad − pb)(br − qc) = (cp − ra)²

If one root is common in two equations.

🔹 Both Roots Common

a/p = b/q = c/r

🔹 Formation of New Equation

If new roots are (α+p) and (β+p)

a(x−p)² + b(x−p) + c = 0

If new roots are (α−p) and (β−p)

a(x+p)² + b(x+p) + c = 0

🔹 Reciprocal Roots

If roots are 1/α , 1/β

ax² + bx + c = 0 → cx² + bx + a = 0

🔹 Nature of Roots

D = 0 → Equal Roots

D > 0 → Real & Unequal

D < 0 → Imaginary

🔹 Conjugate Roots

If one root is α + iβ,

Other root = α − iβ

Roots always occur in pairs.

🔹 Relation Between Roots & Coefficients

Sum = α + β = −b/a

Product = αβ = c/a

|α − β| = √D / |a|

🔹 Cubic Equation

ax³ + bx² + cx + d = 0

If roots are α, β, γ

α+β+γ = −b/a

αβ+βγ+γα = c/a

αβγ = −d/a

🔹 Biquadratic Equation

ax⁴ + bx³ + cx² + dx + e = 0

If roots are α,β,γ,δ

α+β+γ+δ = −b/a

🔹 Important Tip

Always calculate D = b² − 4ac first to know nature of roots.

🔹 Formation of Polynomial

If α₁, α₂, α₃ ... αₙ are roots:

xⁿ − S₁xⁿ⁻¹ + S₂xⁿ⁻² − S₃xⁿ⁻³ + ... = 0

Where S₁, S₂, S₃ are sums of roots.

🔹 Quadratic Formation

If roots are α, β

x² − (α+β)x + αβ = 0

🔹 Cubic Formation

If roots are α, β, γ

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

🔹 Biquadratic Formation

If roots are α,β,γ,δ

x⁴ − S₁x³ + S₂x² − S₃x + S₄ = 0

🔹 Maximum & Minimum Value

For ax² + bx + c

x = −b / 2a

Value = (4ac − b²) / 4a

a > 0 → Min Value a < 0 → Max Value

🔹 Inequality

Expression using > , < , ≥ , ≤

3x + 5 > 7

🔹 Types of Inequality

1. Numerical

5 > 2 , 9 < 11

2. Literal

x + 2 > 7

3. Strict

x > 5

4. Slack

x ≥ 4

🔹 Solution of Inequality

3x + 2 > 11

Subtract 2

3x > 9

Divide by 3

x > 3

🔹 Important Rule

When multiplying or dividing by negative number, sign reverses.

x > 5 → −x < −5

🔹 Exam Tip

Always simplify inequality before solving.

🔹 Polynomial

🔹 Real Polynomial

🔹 Degree of Polynomial

🔹 Types of Polynomial

🔹 Polynomial Equation

🔹 Quadratic Equation

🔹 Roots of Equation

🔹 Factorisation Method

🔹 Direct Formula

🔹 Nature of Roots

🔹 Sridharacharya Formula

🔹 Given Quadratic Equation

🔹 Condition for Common Root

🔹 Both Roots Common

🔹 Formation of New Equation

🔹 Reciprocal Roots

🔹 Nature of Roots

🔹 Conjugate Roots

🔹 Relation Between Roots & Coefficients

🔹 Cubic Equation

🔹 Biquadratic Equation

🔹 Important Tip

🔹 Formation of Polynomial

🔹 Quadratic Formation

🔹 Cubic Formation

🔹 Biquadratic Formation

🔹 Maximum & Minimum Value

🔹 Inequality

🔹 Types of Inequality

🔹 Solution of Inequality

🔹 Important Rule

🔹 Exam Tip

📘 Concept 1: Power Behaviour

📘 Concept 2: Product Maximum

📘 Example

📘 Nature of Roots

📘 Lines Comparison

📘 Sign of Roots

📝 Practice Questions