The GCD and LCM of two polynomials are $x + 1$ and $x^{6} - 1$ respectively. If one of the polynomials is $x^{3} + 1$ , find the other.

Open in App

Solution

Given GCD $= x + 1$ and LCM $= x^{6} - 1$
Let $f (x) = x^{3} + 1$ .
We known that $L C M \times G C D = f (x) \times g (x)$
$\to g (x) = \frac{L C M \times G C D}{f (x)} = \frac{(x^{6} - 1) (x + 1)}{x^{3} + 1}$
$\frac{(x^{3} + 1) (x^{3} - 1) (x + 1)}{x^{3} + 1} = (x^{3} - 1) (x + 1)$
Hence, $g (x) = (x^{3} - 1) (x + 1)$ .

Suggest Corrections

Similar questions

x^{6} - 1

f (x) = x^{3} + 1

Let GCD $= x + 1$ and LCM $= x^{6} - 1$ for two given polynomials $f (x)$ and $g (x)$ . Then, if $f (x) = x^{3} + 1$ . what is $g (x)$ ?

(x^{3} - 4 x) (5 x + 1), (5 x^{2} + x), (5 x^{3} - 9 x^{2} - 2 x)

2 (x + 1) (x^{2} - 4), (x + 1), (x + 1) (x - 2)

x^{3} + x^{2} + x + 1

x^{3} - x^{2} + x - 1

Join BYJU'S Learning Program