Find the GCD ...

Question

Find the GCD of the polynomials $x^{4} + 3 x^{3} - x - 3$ and $x^{3} + x^{2} - 5 x + 3$ .

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Solution

Let $f (x) = x^{4} + 3 x^{3} - x - 3$ and $g (x) = x^{3} + x^{2} - 5 x + 3$
Here degree of f(x) > degree of g(x).
$∴$ Divisor is $x^{3} + x^{2} - 5 x + 3$
Therefore, GCD $(f (x), g (x)) = x^{2} + 2 x - 3$ .

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Determine which of the following polynomials has (x + 1) a factor:

(i) x³ + x² + x + 1 (ii) x⁴ + x³ + x² + x + 1

(iii) x⁴ + 3x³ + 3x² + x + 1 (iv)