Find the LCM:
z^4-14z^3+49z^2,3z^2-147 and 5z+35
z^4-14z^3+49z^2
= z^2(z^2 - 14z+ 49)
= x^2(z-7)^2
3z^2-147
= 3(z^2 - 49)
= 3(z+7)(z-7)
5z + 35
= 5(z+7)
LCM = 15z^2(z+7)(z-7)^2
To find the least common multiple (LCM) of the given algebraic expressions:
Step 1: Factorize each expression completely.
- Factorize z^4 - 14z^3 + 49z^2:
z^4 - 14z^3 + 49z^2 = z^2 (z^2 - 14z + 49) = z^2 (z - 7)^2
- Factorize 3z^2 - 147:
3z^2 - 147 = 3(z^2 - 49) = 3(z - 7)(z + 7)
- Factorize 5z + 35:
5z + 35 = 5(z + 7)
Step 2: Identify the highest power for each factor.
- The highest power of each factor is as follows:
z^2: power = 2
(z - 7): power = 2
(z + 7): power = 1
(z): power = 1
3: power = 1
5: power = 1
Step 3: Take the highest power for each factor and multiply them together.
LCM = z^2 * (z - 7)^2 * (z + 7) * z * 3 * 5
= 3z^5 (z - 7)^2 (z + 7)
Therefore, the LCM of the given expressions is 3z^5 (z - 7)^2 (z + 7).