What is the LCM of #5z^6+30z^5-35z^4# and #7z^7 +98z^6+343z^5#?

Question

Eli Assouline · Accepted Answer

#35z^8+455z^2+1225z-1715#

#5z^6+30z^5-35z^4 = 5z^4(z^2+6z-7) = 5z^4(z+7)(z-1)#

#7z^7+98z^6+343z^5 = 7z^5(z^2+14z+49) = 7z^5(z+7)^2#

So the simplest polynomial which includes all of the factors of these two polynomials in the multiplicities in which they occur is:

#5*7z^5(z+7)^2(z-1) = 35z^5(z^2+14z+49)(z-1)#

#color(white)(5*7z^5(z+7)^2(z-1)) = 35z^5(z^3+(14-1)z^2+(49-14)z-49)#

#color(white)(5*7z^5(z+7)^2(z-1)) = 35z^5(z^3+13z^2+35z-49)#

#color(white)(5*7z^5(z+7)^2(z-1)) = 35z^8+455z^2+1225z-1715#

Nathan Axel · Answer

undefined To find the least common multiple (LCM) of $5z^6 + 30z^5 - 35z^4$ and $7z^7 + 98z^6 + 343z^5$, we need to factor each polynomial and then identify the highest powers of each factor that appear in either expression.

Factoring each polynomial:
$5z^6 + 30z^5 - 35z^4 = 5z^4(z^2 + 6z - 7)$
$7z^7 + 98z^6 + 343z^5 = 7z^5(z^2 + 14z + 49)$

Now, identify the highest powers of each factor:
- For $z^4$: The highest power is $z^4$.
- For $z^5$: The highest power is $z^5$.
- For $z^6$: The highest power is $z^6$.
- For $z^7$: The highest power is $z^7$.
- For the quadratic factors: The highest power of each factor is $z^2$.

Now, multiply these highest powers together:
$(5z^4)(7z^5)(z^6)(z^7)(z^2) = 35z^{24}$

Therefore, the least common multiple of $5z^6 + 30z^5 - 35z^4$ and $7z^7 + 98z^6 + 343z^5$ is $35z^{24}$.