Question

The GCD of ${x^3} - 1$ and ${x^4} - 1$ is
A) ${x^3} - 1$
B) ${x^3} + 1$
C) $x + 1$
D) $x - 1$

Answer 1

Hint:
We will factorize the given polynomials using the algebraic identities. Then we shall consider the common factors amongst the given polynomials and hence find the GCD of the same.

Complete step by step solution:
Let’s consider the polynomial ${x^3} - 1$. Factoring the polynomial using the identity ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$ where $a = x$ and $b = 1$
${x^3} - {1^3} = (x - 1)({x^2} + x + 1)$
So, the factors of ${x^3} - 1$ are $(x - 1)$ and $({x^2} + x + 1)$ .
Now, let’s consider the polynomial ${x^4} - 1$ . Factoring the polynomial using the identity ${a^2} - {b^2} = (a + b)(a - b)$ where $a = {x^2}$ and $b = 1$ .
$
  {x^4} - 1 = {({x^2})^2} - 1 \\
   \Rightarrow {x^4} - 1 = ({x^2} + 1)({x^2} - 1) \\
$
Notice that ${x^2} - 1$ can still be further factorized. So, factorizing ${x^2} - 1$ by using ${a^2} - {b^2} = (a + b)(a - b)$ where $a = x$ and $b = 1$
\[{x^4} - 1 = ({x^2} + 1)(x + 1)(x - 1)\]
So, the factors of ${x^4} - 1$ are \[(x - 1)\] , \[(x + 1)\] and \[({x^2} + 1)\] .
Now, observe that the common factors between ${x^3} - 1$ and ${x^4} - 1$ is \[(x - 1)\] .
Now, we know that the GCD or Greatest Common Divisor is the Greatest among the common factors of the given entities. Since, there is only one common factor, \[(x - 1)\] is the GCD of ${x^4} - 1$ and ${x^3} - 1$ .

Hence, the correct option is D.

Note:
It is imperative to know the factor $({x^2} + x + 1)$ of ${x^3} - 1$ and the factor \[({x^2} + 1)\] of ${x^4} - 1$ can be considered factors of the above respectively only if Imaginary roots of $x$ are considered.