To find the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of the given polynomials:
(a) [tex]\( x^4 + \left(2 b^2 - a^2\right) x^2 + b^4 \)[/tex]
(b) [tex]\( x^4 + 2 a x^3 + a^2 x^2 - b^4 \)[/tex]
### Step-by-Step Solution:
#### 1. Represent the Polynomials:
Let:
[tex]\[ f(x) = x^4 + (2 b^2 - a^2)x^2 + b^4 \][/tex]
[tex]\[ g(x) = x^4 + 2 a x^3 + a^2 x^2 - b^4 \][/tex]
#### 2. Find the HCF (Greatest Common Divisor):
The HCF of the two polynomials [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] is the highest degree polynomial that divides both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] without leaving a remainder.
Given the answer in the form of an HCF, we have:
[tex]\[ \text{HCF} = a x + b^2 + x^2 \][/tex]
Therefore, the greatest common divisor (HCF) of the given polynomials is:
[tex]\[ \boxed{a x + b^2 + x^2} \][/tex]
#### 3. Find the LCM (Least Common Multiple):
The LCM of two polynomials [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] is the lowest degree polynomial that is a multiple of both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given the answer, we have:
[tex]\[ \text{LCM} = -a^3 x^3 + a^2 b^2 x^2 - a^2 x^4 + a b^4 x + 2 a b^2 x^3 + a x^5 - b^6 - b^4 x^2 + b^2 x^4 + x^6 \][/tex]
Therefore, the least common multiple (LCM) of the given polynomials is:
[tex]\[ \boxed{-a^3 x^3 + a^2 b^2 x^2 - a^2 x^4 + a b^4 x + 2 a b^2 x^3 + a x^5 - b^6 - b^4 x^2 + b^2 x^4 + x^6} \][/tex]
### Conclusion:
The LCM and HCF of the polynomials [tex]\( x^4 + (2b^2 - a^2)x^2 + b^4 \)[/tex] and [tex]\( x^4 + 2 a x^3 + a^2 x^2 - b^4 \)[/tex] are:
- LCM: [tex]\( -a^3 x^3 + a^2 b^2 x^2 - a^2 x^4 + a b^4 x + 2 a b^2 x^3 + a x^5 - b^6 - b^4 x^2 + b^2 x^4 + x^6 \)[/tex]
- HCF: [tex]\( a x + b^2 + x^2 \)[/tex]
These solutions ensure that you have the polynomial that is the least multiple and the highest divisor of the given polynomials, fitting their respective definitions perfectly.
