The GCD of $x^{4} + 3 x^{3} + 5 x^{2} + 26 x + 56$ and $x^{4} + 2 x^{3} - 4 x^{2} - x + 28$ is $x^{2} + 5 x + 7$ .
Find their LCM.

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Solution

Let $f (x) = x^{4} + 3 x^{3} + 5 x^{2} + 26 x$ and $g (x) = x^{4} + 2 x^{3} - 4 x^{2} - x + 28$
Given that GCD $= x^{2} + 5 x + 7$ . Also, we have GCD $\times$ LCM= f(x) $\times$ g(x).
Thus, $L C M = \frac{f (x) \times g (x)}{G C D}$
Now, GCD divides both f(x) and g(x).
Let us divided f(X) by the GCD.
When f(x) is divided by GCD, we get the quoteint as $x^{2} - 2 x + 8$ .
Now, (1) $\to$ LCM $= (x^{2} - 2 x + 8) \times g (x)$
Thus, LCM $= (x^{2} - 2 x + 8) (x^{4} + 2 x^{3} - 4 x^{2} - x + 28)$ .

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