To find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of the given polynomials:
[tex]\[ f(x) = x^4 + (2b^2 - a^2)x^2 + b^4 \][/tex]
[tex]\[ g(x) = x^4 + 2ax^3 + a^2x^2 - b^4 \][/tex]
we will perform the following steps:
### 1. Determine the Greatest Common Divisor (GCD) or Highest Common Factor (HCF)
The Greatest Common Divisor (GCD) of two polynomials is the highest degree polynomial that divides both without leaving a remainder. Given the polynomials:
[tex]\[ f(x) = x^4 + (2b^2 - a^2)x^2 + b^4 \][/tex]
[tex]\[ g(x) = x^4 + 2ax^3 + a^2x^2 - b^4 \][/tex]
The GCD is:
[tex]\[ \text{GCD}(f(x), g(x)) = x^2 + ax + b^2. \][/tex]
### 2. Determine the Least Common Multiple (LCM)
The Least Common Multiple (LCM) of the polynomials is determined using the relationship:
[tex]\[ \text{LCM}(f(x), g(x)) = \frac{f(x) \cdot g(x)}{\text{GCD}(f(x), g(x))} \][/tex]
So we first multiply the given polynomials:
[tex]\[ f(x) \cdot g(x) = (x^4 + (2b^2 - a^2)x^2 + b^4) \cdot (x^4 + 2ax^3 + a^2x^2 - b^4) \][/tex]
Then we divide by the GCD:
[tex]\[ \text{LCM}(f(x), g(x)) = \frac{(x^4 + (2b^2 - a^2)x^2 + b^4) \cdot (x^4 + 2ax^3 + a^2x^2 - b^4)}{x^2 + ax + b^2} \][/tex]
The result simplifies to:
[tex]\[ \text{LCM}(f(x), g(x)) = x^6 + x^5a - x^4a^2 + x^4b^2 - x^3a^3 + 2x^3ab^2 + x^2a^2b^2 - x^2b^4 + xab^4 - b^6. \][/tex]
### Summary
Given the polynomials:
[tex]\[ f(x) = x^4 + (2b^2 - a^2)x^2 + b^4 \][/tex]
[tex]\[ g(x) = x^4 + 2ax^3 + a^2x^2 - b^4 \][/tex]
we have:
Greatest Common Divisor (GCD) or Highest Common Factor (HCF):
[tex]\[ \text{GCD}(f(x), g(x)) = x^2 + ax + b^2. \][/tex]
Least Common Multiple (LCM):
[tex]\[ \text{LCM}(f(x), g(x)) = x^6 + x^5a - x^4a^2 + x^4b^2 - x^3a^3 + 2x^3ab^2 + x^2a^2b^2 - x^2b^4 + xab^4 - b^6. \][/tex]
