Finding the Least Common Multiple of Monomials and Polynomials
To find the least common multiple (LCM) of the given monomials \( 5x^7 \), \( 5x^6 - 10x^5 \), and \( 2x^8 + 8x^7 + 8x^6 \), you would first factor each monomial or polynomial into prime factors and include variables with their highest powers.
Let's start by factoring the given expressions:
1. \( 5x^7 \) is already in its factored form.
2. \( 5x^6 - 10x^5 \) can be factored as \( 5x^5(x - 2) \).
3. \( 2x^8 + 8x^7 + 8x^6 \) can be factored by pulling out the greatest common factor, which is \( 2x^6 \), leaving us with \( 2x^6(x^2 + 4x + 4) \). However, since we are only interested in the LCM, which concerns the greatest powers of variables and prime factors, we don't need to factor the quadratic expression further.
Now, we take the highest powers of each prime factor found in the terms:
- The highest power of 2 is \( 2^1 \) from \( 2x^6(x^2 + 4x + 4) \).
- The highest power of 5 is \( 5^1 \) from \( 5x^7 \) and \( 5x^5(x - 2) \).
- The highest power of \( x \) is \( x^8 \) from \( 2x^8 + 8x^7 + 8x^6 \).
Combining these, the LCM of the given expressions is:
\[ 2^1 \cdot 5^1 \cdot x^8 = 10x^8 \]
Therefore, the LCM is \( 10x^8 \).
