Chapter A.3, Problem 138AYU

To determine

To find: The expression $x^2+x+1$ is prime.

Answer to Problem 138AYU

The expression $x^2+x+1$ must be prime.

Explanation of Solution

Given information:

The expression is $x^2+4$ .

Calculation:

Calculate the polynomial to be prime, it should be able to written as a product of the form $\left(x+a\right)\left(x+b\right)$ . The value of $a$ and $b$ indicate the values of $x$ that cause the original quadratic equation to be equal to $0$ .

If,

  $\begin{array}{l}{x}^2+4=0\\ x^2=-4\end{array}$

We see that the square of a number is equal to negative number, which can never be possible for real values of $x$ . The square of a real number is always positive. This shows that $x^2+4$ is prime.

Consider when $a$ and $b$ are 1

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

Since this doesn’t equal $x^2+x+1$ when multiply out, these factors are not an option.

Consider when $a$ and $b$ are -1

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

Since this doesn’t equal $x^2+x+1$ when multiplied out, these factors are not option either.

This means that $x^2+x+1$ must be prime