To find the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of the given polynomials
[tex]\[ P_1(x) = x^4 + (2b^2 - a^2)x^2 + b^4 \][/tex]
and
[tex]\[ P_2(x) = x^4 + 2ax^3 + a^2x^2 - b^4, \][/tex]
we proceed as follows:
### Step-by-Step Solution
#### 1. Understand the Polynomials:
We are given two polynomials in terms of [tex]\(x\)[/tex], [tex]\(a\)[/tex], and [tex]\(b\)[/tex]. The aim is to find the common factors and the least common multiple of these polynomials.
#### 2. Identify Patterns and Common Factors:
- The first polynomial:
[tex]\[ P_1(x) = x^4 + (2b^2 - a^2)x^2 + b^4 \][/tex]
- The second polynomial:
[tex]\[ P_2(x) = x^4 + 2ax^3 + a^2x^2 - b^4 \][/tex]
These polynomials are both quartic (degree [tex]\(4\)[/tex]).
#### 3. Compute the Highest Common Factor (HCF):
The HCF of two polynomials is the polynomial of the highest degree that divides both polynomials without leaving a remainder.
For the given polynomials, the Highest Common Factor (HCF) is:
[tex]\[ HCF(P_1(x), P_2(x)) = a x + b^2 + x^2 \][/tex]
#### 4. Compute the Least Common Multiple (LCM):
The LCM of two polynomials is the polynomial of the lowest degree that both polynomials can divide without leaving a remainder.
For the given polynomials, the Least Common Multiple (LCM) is:
[tex]\[ LCM(P_1(x), P_2(x)) = -a^3x^3 + a^2b^2x^2 - a^2x^4 + a b^4x + 2a b^2 x^3 + a x^5 - b^6 - b^4 x^2 + b^2 x^4 + x^6 \][/tex]
### Final Answer:
- The Highest Common Factor (HCF) of the polynomials is:
[tex]\[ a x + b^2 + x^2 \][/tex]
- The Least Common Multiple (LCM) of the polynomials is:
[tex]\[ -a^3x^3 + a^2b^2x^2 - a^2x^4 + a b^4x + 2a b^2 x^3 + a x^5 - b^6 - b^4 x^2 + b^2 x^4 + x^6 \][/tex]
These are the HCF and LCM for the given polynomials.
