Find the L.C.M of the polynomials $90 (x^{2} - 5 x + 6) {(2 x + 1)}^{2}$ and $140 {(x - 3)}^{2} (2 x^{2} + 15 x + 7)$

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Solution

Let $p (x) = 90 (x^{2} - 5 x + 6) {(2 x + 1)}^{2}$

$= 5 \times 2 \times 3 \times 3 (x^{2} - 2 x - 3 x + 6) {(2 x + 1)}^{2}$

$= 5 \times 2 \times 3 \times 3 (x (x - 2) - 3 (x - 2)) {(2 x + 1)}^{2}$

$= 5 \times 2 \times 3 \times 3 (x - 2) (x - 3) {(2 x + 1)}^{2}$

Let $q (x) = 140 {(x - 3)}^{2} (2 x^{2} + 15 x + 7)$

$= 2 \times 2 \times 5 \times 7 {(x - 3)}^{2} (2 x^{2} + 14 x + x + 7)$

$= 2 \times 2 \times 5 \times 7 {(x - 3)}^{2} (2 x (x + 7) + (x + 7))$

$= 2 \times 2 \times 5 \times 7 {(x - 3)}^{2} (x + 7) (2 x + 1)$

L.C.M of $p (x)$ and $q (x)$ is

$= 2^{2} \times 3^{2} \times 5 \times 7 (x - 2) {(x - 3)}^{2} {(2 x + 1)}^{2} (x + 7)$

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