L.C.M And G.C.D of Polynomials

Polynomials

expression a₀xⁿ+a₁x^n-1+a₂x^n-2+a₃x^n-3+.........+ a_n-1x+a_n
where a₀,a₁,a₂,a₃.....a_n are real number and n is non negative integer and a₀ not equal to zero is polynomial of degree n.

a₀,a₁,a₂,a₃.....a_n are coefficients of polynomial.
if coefficients of polynomial are integer then polynomial are written as
p(x)=a₀xⁿ+a₁x^n-1+a₂x^n-2+a₃x^n-3+.........+ a_n-1x+a_n


Example:

5x-2 is a polynomial of degree 1
5x²-2x+4 is a polynomial of degree 2
x³+3x²-2x+6 is a polynomial of degree 3

What is Divisor or Factor

A Polynomial d(x) is divisior of a Polynomial p(x) if d(x) is a factor of p(x). For example p(x)=d(x)r(x)

In this example d(x) is factor of p(x) so we called d(x) is divisor(factor) of Polynomial p(x).

Example

p(x)=x²-5x+6 then p(x)=(x-3)(x-2)

here (x-3) and (x-2) is a factor of p(x) so (x-3) and (x-2) are divisors or factors of polynomial p(x).

Greatest common divisor(G.C.D) or HCF of Polynomials

Greatest common divisor(g.c.d) or hcf of two polynomials are highest degree common divisor of both polynomials. and coefficient of highest degree term is positive.

If p(x) and q(x) are two polynomials then highest degree common divisor of p(x) and q(x) is called as greatest common divisor(g.c.d) or hcf of Polynomials.

Example
G.C.D of polynomials 2x²-x-1 and 4x²+8x+3
p(x)=2x²-x-1
p(x)=2x²-2x+x-1
p(x)=(2x+1)(x-1) here (2x+1) and (x-1) is factor of polynomial p(x)

second polynomial
q(x)=4x²+8x+3
q(x)=4x²+6x+2x+3
q(x)=2x(2x+3)+1(2x+3)
q(x)=(2x+1)(2x+3) here (2x+1) and (2x+3) is factor of polynomial q(x)


in both polynomial only (2x+1) is only common divisor with least exponent 1
so GCD of p(x) and q(x) is (2x+1)¹=(2x+1)

Least Common Multiple(LCM) of Polynomials

Polynomial with the lowest degree and having smallest numerical coefficient which is divisible by the given polynomials and cofficient of highest degree term has the same sign as the cofficient of highest degree term in ther product is called Least Common Multiple(LCM) of two or more Polynomials

How to find L.C.M of Polynomials
1. Find Factors of given polynomials and write them as a product of power of irreducible factors.

2. List all irreducible factors and find greatest exponent in the factorized form .
3. Raise each irreducible factors to the greatest exponent and multiply them.
4. Get LCM of polynomials

Example:

How to find LCM of Polynomials f(x)=18x⁴-36x³+18x² and g(x)=45x⁶-45x³

First polynomial f(x)=18x⁴-36x³+18x²
f(x)=18x²(x²-2x+1)
f(x)=18x²(x-1)²
f(x)=2*3²*x²*(x-1)²

second polynomial g(x)=45x⁶-45x³

g(x)=45x³(x³-1)
g(x)=45x³(x-1)(x²+x+1)
g(x)=3²*5*x³*(x-1)(x²+x+1)

In both polynomials irreducible factors are 2,3,5,x,x-1 and x²+x+1
Highest component are 1,2,1,3,2,1

So LCM =2¹*3²*5¹*x³*(x-1)²*(x²+x+1)

LCM =90x³(x-1)²(x²+x+1)