Theory of Equations
and Inequations
Polynomial
An algebraic expression of the form a0 + a1x + a2x 2 +...+ an x n , where
n ∈ N , is called a polynomial. It is generally denoted by p( x ), q( x ),
f ( x ), g( x ) etc.
Real Polynomial
Let a0 , a1 , a2 , K , an be real numbers and x is a real variable, then,
f ( x ) = a0 + a1x + a2x 2 + K + an x n is called a real polynomial of real
variable x with real coefficients.
Complex Polynomial
If a0 , a1 , a2 , K , an be complex numbers and x is a varying complex
number, then f ( x ) = a0 + a1x + a2x 2 + K + an − 1x n − 1 + an x n is called a
complex polynomial or a polynomial of complex variable x with complex
coefficients.
Degree of a Polynomial
A polynomial f ( x ) = a0 + a1x + a2x 2 + a3 x3 + K + an x n , real or complex
is a polynomial of degree n, if an ≠ 0.
Some Important Deduction
(i) Linear Polynomial A polynomial of degree one is known as
linear polynomial.
(ii) Quadratic Polynomial A polynomial of second degree is
known as quadratic polynomial.
(iii) Cubic Polynomial A polynomial of degree three is known as
cubic polynomial.
(iv) Biquadratic Polynomial A polynomial of degree four is
known as biquadratic polynomial.
,Polynomial Equation
If f ( x ) is a polynomial, real or complex, then f ( x ) = 0 is called a
polynomial equation.
Quadratic Equation
A quadratic polynomial f ( x ) when equated to zero is called quadratic
equation.
i.e. ax 2 + bx + c = 0, where a , b, c ∈ R and a ≠ 0.
Roots of a Quadratic Equation
The values of variable x which satisfy the quadratic equation is called
roots of quadratic equation.
Solution of Quadratic Equation
1. Factorisation Method
Let ax 2 + bx + c = a( x − α ) ( x − β ) = 0. Then, x = α and x = β will satisfy
the given equation.
2. Direct Formula
Quadratic equation ax 2 + bx + c = 0 ( a ≠ 0) has two roots, given by
− b + b2 − 4ac − b − b2 − 4ac
α= ,β =
2a 2a
− b+ D − b− D
or α= ,β =
2a 2a
where, D = ∆ = b2 − 4ac is called discriminant of the equation.
Above formulas also known as Sridharacharya formula.
Nature of Roots
(i) Let quadratic equation be ax 2 + bx + c = 0, whose discriminant
is D.
Also, let a , b , c ∈ R and a ≠ 0. Then,
(a) D < 0 ⇒ Complex roots
(b) D > 0 ⇒ Real and distinct roots
b
(c) D = 0 ⇒ Real and equal roots as α = β = −
2a
, Note If a, b, c ∈ Q, a ≠ 0, then
(a) D > 0 and D is a perfect square.
⇒ Roots are unequal and rational.
(b) D > 0, a = 1; b, c ∈ I and D is a perfect square.
⇒ Roots are integral.
(c) D > 0 and D is not a perfect square.
⇒ Roots are irrational and unequal.
(ii) Conjugate Roots The irrational (complex) roots of a quadratic
equation, whose coefficients are rational (real) always occur in
conjugate pairs. Thus,
(a) if one root be α + iβ, then other root will be α − iβ.
(b) if one root be α + β , then other root will be α − β .
(iii) Let D1 and D2 are the discriminants of two quadratic equations.
(a) If D1 + D2 ≥ 0, then atleast one of D1 and D2 ≥ 0
Thus, if D1 < 0, then D2 > 0, if D2 < 0, then D1 > 0 or D1 and
D2 both can be non-negative (means positive or zero).
(b) If D1 + D2 < 0, then atleast one of D1 and D2 < 0
Thus, if D1 > 0, then D2 < 0, if D2 > 0, then D1 < 0 or D1 and
D2 both can be negative.
Roots Under Particular Conditions
For the quadratic equation ax 2 + bx + c = 0.
(i) If a > 0 and b = 0, roots are real/complex according as c < 0 or c > 0.
These roots are equal in magnitude but of opposite sign.
(ii) If c = 0, one root is zero, other is − b / a.
(iii) If b = c = 0, both roots are zero.
(iv) If a = c, roots are reciprocal to each other.
a > 0, c < 0
(v) If ⇒ Roots are of opposite sign.
a < 0, c > 0
a > 0, b > 0, c > 0
(vi) If ⇒ Both roots are negative, provided D ≥ 0
a < 0, b < 0, c < 0
a > 0, b < 0, c > 0
(vii) If ⇒ Both roots are positive, provided D ≥ 0
a < 0, b > 0, c < 0
(viii) If sign of a = sign of b ≠ sign of c
⇒ Greater root in magnitude is negative.
(ix) If sign of b = sign of c ≠ sign of a
⇒ Greater root in magnitude is positive.
(x) If a + b + c = 0 ⇒ One root is 1 and second root is c/a.
and Inequations
Polynomial
An algebraic expression of the form a0 + a1x + a2x 2 +...+ an x n , where
n ∈ N , is called a polynomial. It is generally denoted by p( x ), q( x ),
f ( x ), g( x ) etc.
Real Polynomial
Let a0 , a1 , a2 , K , an be real numbers and x is a real variable, then,
f ( x ) = a0 + a1x + a2x 2 + K + an x n is called a real polynomial of real
variable x with real coefficients.
Complex Polynomial
If a0 , a1 , a2 , K , an be complex numbers and x is a varying complex
number, then f ( x ) = a0 + a1x + a2x 2 + K + an − 1x n − 1 + an x n is called a
complex polynomial or a polynomial of complex variable x with complex
coefficients.
Degree of a Polynomial
A polynomial f ( x ) = a0 + a1x + a2x 2 + a3 x3 + K + an x n , real or complex
is a polynomial of degree n, if an ≠ 0.
Some Important Deduction
(i) Linear Polynomial A polynomial of degree one is known as
linear polynomial.
(ii) Quadratic Polynomial A polynomial of second degree is
known as quadratic polynomial.
(iii) Cubic Polynomial A polynomial of degree three is known as
cubic polynomial.
(iv) Biquadratic Polynomial A polynomial of degree four is
known as biquadratic polynomial.
,Polynomial Equation
If f ( x ) is a polynomial, real or complex, then f ( x ) = 0 is called a
polynomial equation.
Quadratic Equation
A quadratic polynomial f ( x ) when equated to zero is called quadratic
equation.
i.e. ax 2 + bx + c = 0, where a , b, c ∈ R and a ≠ 0.
Roots of a Quadratic Equation
The values of variable x which satisfy the quadratic equation is called
roots of quadratic equation.
Solution of Quadratic Equation
1. Factorisation Method
Let ax 2 + bx + c = a( x − α ) ( x − β ) = 0. Then, x = α and x = β will satisfy
the given equation.
2. Direct Formula
Quadratic equation ax 2 + bx + c = 0 ( a ≠ 0) has two roots, given by
− b + b2 − 4ac − b − b2 − 4ac
α= ,β =
2a 2a
− b+ D − b− D
or α= ,β =
2a 2a
where, D = ∆ = b2 − 4ac is called discriminant of the equation.
Above formulas also known as Sridharacharya formula.
Nature of Roots
(i) Let quadratic equation be ax 2 + bx + c = 0, whose discriminant
is D.
Also, let a , b , c ∈ R and a ≠ 0. Then,
(a) D < 0 ⇒ Complex roots
(b) D > 0 ⇒ Real and distinct roots
b
(c) D = 0 ⇒ Real and equal roots as α = β = −
2a
, Note If a, b, c ∈ Q, a ≠ 0, then
(a) D > 0 and D is a perfect square.
⇒ Roots are unequal and rational.
(b) D > 0, a = 1; b, c ∈ I and D is a perfect square.
⇒ Roots are integral.
(c) D > 0 and D is not a perfect square.
⇒ Roots are irrational and unequal.
(ii) Conjugate Roots The irrational (complex) roots of a quadratic
equation, whose coefficients are rational (real) always occur in
conjugate pairs. Thus,
(a) if one root be α + iβ, then other root will be α − iβ.
(b) if one root be α + β , then other root will be α − β .
(iii) Let D1 and D2 are the discriminants of two quadratic equations.
(a) If D1 + D2 ≥ 0, then atleast one of D1 and D2 ≥ 0
Thus, if D1 < 0, then D2 > 0, if D2 < 0, then D1 > 0 or D1 and
D2 both can be non-negative (means positive or zero).
(b) If D1 + D2 < 0, then atleast one of D1 and D2 < 0
Thus, if D1 > 0, then D2 < 0, if D2 > 0, then D1 < 0 or D1 and
D2 both can be negative.
Roots Under Particular Conditions
For the quadratic equation ax 2 + bx + c = 0.
(i) If a > 0 and b = 0, roots are real/complex according as c < 0 or c > 0.
These roots are equal in magnitude but of opposite sign.
(ii) If c = 0, one root is zero, other is − b / a.
(iii) If b = c = 0, both roots are zero.
(iv) If a = c, roots are reciprocal to each other.
a > 0, c < 0
(v) If ⇒ Roots are of opposite sign.
a < 0, c > 0
a > 0, b > 0, c > 0
(vi) If ⇒ Both roots are negative, provided D ≥ 0
a < 0, b < 0, c < 0
a > 0, b < 0, c > 0
(vii) If ⇒ Both roots are positive, provided D ≥ 0
a < 0, b > 0, c < 0
(viii) If sign of a = sign of b ≠ sign of c
⇒ Greater root in magnitude is negative.
(ix) If sign of b = sign of c ≠ sign of a
⇒ Greater root in magnitude is positive.
(x) If a + b + c = 0 ⇒ One root is 1 and second root is c/a.
