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Complex numbers are defined as expressions of the form a + ib where a, b ∊ R & . It is denoted by z i.e. z=a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z).
EVERY COMPLEX NUMBER CAN BE REGARDED AS
If b = 0
If a = 0
If b ≠ 0
ALGEBRA OF COMPLEX NUMBER
The algebraic operations on complex numbers are similar to those on real numbers treating ‘I’ as a polynomial. In equalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.
e.g. z > 0, 4 + 2i < 2+4i are meaningless.
However in real numbers if a2 + b2=0 then a = 0 =b
but in complex numbers,
z12+z22=0 does not imply z1=z2=0
Equality In Complex Number
Two complex numbers z1= a1 + ib1 & z2 =a2 +ib2 are equal if and only if their real & imaginary parts coincide.
If z=a + ib then its conjugate complex is obtained by changing the sign of its imaginary & is denoted by
A function f defined by f(x) = anxn+an-1xn-1+……..+a1x+a0, where a0, a1, a2….. an ∊ R is called n degree polynomial while coefficient (an ≠0, n ∊ W) is real .If a0 , a2 ….an ∊ C, then it is called complex coefficient polynomial.
A polynomial of degree two in one variable f(x) = y = ax2 + bx + C, where a ≠ 0 & a, b, c ∊ R
a → leading coefficient, c → absolute term/ constant term
If a = 0 then y = bx + c → linear polynomial b ≠ 0
If a = 0, c = 0 then y = bx → odd linear polynomial
1. The solution of the quadratic equation, ax2+ bx + c = 0 is given by
The expression b2-4ac = D is called the discriminant of the quadratic equation.
2. If α & β are the roots of the quadratic equation
Ax2 + bx + c =0 then,
NATURE OF ROOTS
1. Consider the quadratic equation ax2 + bx + c = 0 where a, b, c ∊ R, a ≠ 0 then;
2. Consider the quadratic equation ax2+ bx + c = 0 where a, b, c ∊ Q & a≠0 then;
GRAPH OF QUADRATIC EXPRESSION
RELATION BETWEEN ROOTS & COEFICIENTS
A quadratic equation whose roots are α & β is
(x – α)(x - β)=0
i.e. x2-(α + β)X +αβ=0
i.e. x2-(sum of roots)x + product of roots=0
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