Question 29

What is the GCD of these polynomials? $$(x^{3} + x^{2}+ x + 1)$$ and $$(x^{3} + 2x^{2} + x + 2)$$?

Solution

$$(x^3+x^2+x+1)$$

$$=x^2\left(x+1\right)+1\left(x+1\right).$$

$$=\left(x^2+1\right)\left(x+1\right).$$

And,

$$(x^3+2x^2+x+2)$$

$$=x^2\left(x+2\right)+1\left(x+2\right).$$

$$=\left(x^2+1\right)\left(x+2\right).$$

So, Greatest common divisor between these two is $$\left(x^2+1\right).$$

C is correct choice.

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