One more:
The LCM of f(x) = 4(x - 1)^2 (x^2 + 6x + 5) and g(x) = 10(x - 1) (x + 2) * (x^2 + 7x + 10) is ...
To find the least common multiple (LCM) of the polynomials f(x) and g(x), we need to first factorize each polynomial completely. Then, we can determine the LCM by taking the highest power of each factor.
Let's start by factoring each polynomial:
f(x) = 4(x - 1)^2 (x^2 + 6x + 5)
g(x) = 10(x - 1) (x + 2) (x^2 + 7x + 10)
Now, let's break down the factors of each polynomial:
f(x) = 4(x - 1)(x - 1) (x + 1)(x + 5)
g(x) = 10(x - 1) (x + 2) (x + 5)(x + 2)
To find the LCM, we need to consider each factor and take the highest power:
(x - 1) occurs in both f(x) and g(x), and its highest power is 1.
(x + 1) occurs only in f(x), so its highest power is 1.
(x + 2) occurs in both f(x) and g(x), and its highest power is 2.
(x + 5) occurs in both f(x) and g(x), and its highest power is 1.
Now, let's piece together the LCM using the highest powers of each factor:
LCM = (x - 1)^2 (x + 1) (x + 2)^2 (x + 5)
Therefore, the LCM of f(x) and g(x) is (x - 1)^2 (x + 1) (x + 2)^2 (x + 5).