Certainly! Let's find the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of the given polynomials [tex]\( P(x) \)[/tex] and [tex]\( a(x) \)[/tex].
First, we are given the polynomials:
[tex]\[ P(x) = 5x^3 + 9x^2 + 7x \][/tex]
[tex]\[ a(x) = 5x^2 + 6x \][/tex]
### Finding the Highest Common Factor (HCF)
The HCF of two polynomials is the greatest polynomial that divides both of them.
1. Factorize the polynomials as much as possible:
[tex]\[ P(x) = 5x^3 + 9x^2 + 7x = x(5x^2 + 9x + 7) \][/tex]
[tex]\[ a(x) = 5x^2 + 6x = x(5x + 6) \][/tex]
2. Identify common factors:
Both polynomials contain a common factor of [tex]\( x \)[/tex].
Therefore, the HCF of [tex]\( P(x) \)[/tex] and [tex]\( a(x) \)[/tex] is:
[tex]\[ \text{HCF}(P(x), a(x)) = x \][/tex]
### Finding the Least Common Multiple (LCM)
The LCM of two polynomials is the smallest polynomial that is divisible by both of them.
1. Use the relationship between HCF and LCM:
The product of the HCF and LCM of two polynomials is equal to the product of the polynomials themselves:
[tex]\[ \text{LCM}(P(x), a(x)) \cdot \text{HCF}(P(x), a(x)) = P(x) \cdot a(x) \][/tex]
2. Substitute the known HCF into the equation:
[tex]\[ \text{LCM}(P(x), a(x)) \cdot x = (5x^3 + 9x^2 + 7x) \cdot (5x^2 + 6x) \][/tex]
3. Calculate [tex]\( P(x) \cdot a(x) \)[/tex]:
Let's multiply the polynomials:
[tex]\[ (5x^3 + 9x^2 + 7x)(5x^2 + 6x) \][/tex]
Expand this product:
[tex]\[ = (5x^3)(5x^2) + (5x^3)(6x) + (9x^2)(5x^2) + (9x^2)(6x) + (7x)(5x^2) + (7x)(6x) \][/tex]
[tex]\[ = 25x^5 + 30x^4 + 45x^4 + 54x^3 + 35x^3 + 42x^2 \][/tex]
Combine like terms:
[tex]\[ = 25x^5 + 75x^4 + 89x^3 + 42x^2 \][/tex]
4. Solve for [tex]\(\text{LCM}(P(x), a(x))\)[/tex]:
[tex]\[ \text{LCM}(P(x), a(x)) \cdot x = 25x^5 + 75x^4 + 89x^3 + 42x^2 \][/tex]
Divide both sides by [tex]\( x \)[/tex]:
[tex]\[ \text{LCM}(P(x), a(x)) = \frac{25x^5 + 75x^4 + 89x^3 + 42x^2}{x} \][/tex]
[tex]\[ \text{LCM}(P(x), a(x)) = 25x^4 + 75x^3 + 89x^2 + 42x \][/tex]
Therefore:
[tex]\[ \text{LCM}(P(x), a(x)) = 25x^4 + 75x^3 + 89x^2 + 42x \][/tex]
### Final Answer
The LCM and HCF of the given polynomials [tex]\( P(x) = 9x^2 + 7x + 5x^3 \)[/tex] and [tex]\( a(x) = 5x^2 + 6x \)[/tex] are:
- [tex]\(\text{LCM}(P(x), a(x)) = 25x^4 + 75x^3 + 89x^2 + 42x\)[/tex]
- [tex]\(\text{HCF}(P(x), a(x)) = x\)[/tex]
These values provide the LCM and HCF for the given polynomials accurately.
