Question

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

Answer 1

LCM is $(x^2-2x+8)(x^4+2x^3-4x^2-x+28)$.

Step-by-step explanation:

Given:

GCD of $x^4+3x^3+5x^2+26x+56$ and $x^4+2x^3-x+28$ is $x^2+5x+7$.

Let f(x)= $x^4+3x^3+5x^2+26x + 56$ and g(x) =$x^4+2x^3-4x^2-x+28$

We have GCD = $x^2+5x+7$

Also, we have $GCD \times LCM= f(x) \times g(x).$

Thus, LCM = $\frac{f(x)\times g(x)}{GCD}$

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 $x^2-2x+8x$  [shown in the figure below]  

Now, (1) ⇒ LCM = $(x^2-2x+8) \times g(x)$

Thus, LCM = $(x^2-2x+8)(x^4+2x^3-4x^2-x+28)$