How do you find the LCM of 5x ^5 +10x^4-15x^3 and 3x ^ 8+ 18x^ 7+ 27x^ 6?

1 Answer
Oct 28, 2016

LCM is $15 {x}^{6} \left(x - 1\right) {\left(x + 3\right)}^{2}$

Explanation:

For finding the LCM of two polynomials we should first factorize them.

$5 {x}^{5} + 10 {x}^{4} - 15 {x}^{3}$

= $5 {x}^{3} \left({x}^{2} + 2 x - 3\right) = 5 {x}^{3} \left({x}^{2} + 3 x - x - 3\right)$

= $5 {x}^{3} \left(x \left(x + 3\right) - 1 \left(x + 3\right)\right) = 5 {x}^{3} \left(x - 1\right) \left(x + 3\right)$

$3 {x}^{8} + 18 {x}^{7} + 27 {x}^{6}$

= $3 {x}^{6} \left({x}^{2} + 6 x + 9\right) = 3 {x}^{6} \left({x}^{2} + 3 x + 3 x + 9\right)$

= $3 {x}^{6} \left(x \left(x + 3\right) + 3 \left(x + 3\right)\right) = 5 {x}^{3} \left(x + 3\right) \left(x + 3\right)$

For LCM we should bring all the factors of the two polynomials together, but for common factors we should choose the highest power in either of the two polynomials. Just as we do for finding LCM of numbers,

Hence, LCM is $3 \times 5 \times {x}^{6} \left(x - 1\right) {\left(x + 3\right)}^{2}$ or $15 {x}^{6} \left(x - 1\right) {\left(x + 3\right)}^{2}$