Question

State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence.

$\underset{¯}{x^2+5x+6}$is an example of a prime polynomial.

Short answer

The given sentence is false.The true sentence is${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{5}\mathit{x}\mathbf{+}\mathbf{3}$is an example of a prime polynomial.

Step-by-step solution

Step 1. State whether the given sentence is true or false.

The given expression is $x^2+5x+6$.

It can be written as:

$x^2+5x+6=x^2+2x+3x+6$

$$\begin{gathered} =x\left(x+2\right)+3\left(x+2\right) \\ =\left(x+2\right)\left(x+3\right) \end{gathered}$$

Therefore, it can be noticed that the factors of $x^2+5x+6$ are $x+2$, $x+3$.

Therefore, the expression $x^2+5x+6$ have factors as role="math" localid="1647522371146" $x+2$ and role="math" localid="1647522378415" $x+3$ other than one and itself.

The prime polynomial is a polynomial that has factors as 1 and itself but the polynomial $x^2+5x+6$ have factors as $x+2$ and $x+3$ other than one and itself.

Therefore, the polynomial $x^2+5x+6$ is not a prime polynomial.

Therefore, the given sentence is false.

Step 2. Replace the underlined phrase or expression to make a true sentence.

Consider the polynomial$x^2+5x+3$.

It can be seen that $x^2+5x+3$ has no other factor than 1 and itself. Therefore, the polynomial $x^2+5x+3$ is a prime polynomial.

Therefore, $x^2+5x+3$ is an example of a prime polynomial.