the LCM and HCF of polynomial P(x) and Q(x) are $$56(x^4 + x)$$ and $$4(x^2 -x + 1)$$, respectively. If the polynomial P(x) = $$28(x^3 + 1),$$ then Q(x) is equal to​

Correct option is C

Given:

LCM of P(x) and Q(x) = 56(x^4 + x)

HCF of P(x) and Q(x) is 4(x^2 - x + 1)

P(x) = 28(x^3 + 1)

We are to find Q(x)

Concept Used:
For any two polynomials:

​$$\text{LCM}(P(x), Q(x)) \cdot \text{HCF}(P(x), Q(x)) = P(x) \cdot Q(x)$$​​

Solution:

LCM = $$56(x^4 + x)$$​​

HCF = $$4(x^2 - x + 1)$$​​

P(x) = $$28(x^3 + 1)$$ 

Using the formula;

​$$Q(x) = \frac{56(x^4 + x) \cdot 4(x^2 - x + 1)}{28(x^3 + 1)}$$​ 

Q(x) = $$8 \cdot \frac{x(x + 1)(x^2 - x + 1) \cdot (x^2 - x + 1)}{(x + 1)(x^2 - x + 1)}$$​​

Q(x) $$= 8x(x^2 - x + 1)$$​​