Question

Which of the following polynomials is prime?

A$\mathbf{5}{x}^2+\mathbf{34}x+\mathbf{24}$

B$\mathbf{4}{x}^2+\mathbf{22}x+\mathbf{10}$

C$\mathbf{4}{x}^2+\mathbf{38}x+\mathbf{70}$

D$\mathbf{5}{x}^2+\mathbf{3}x+\mathbf{4}$

Short answer

$\mathbf{5}{x}^2+\mathbf{3}x+\mathbf{4}$

Step-by-step solution

Step 1. Factor 5x2+34x+24.

Factorized the polynomial, $5x^2+34x+24$.

$$\begin{gathered} 5x^2+34x+24=5x^2+30x+4x+24 \\ =5x\left(x+6\right)+4\left(x+6\right) \\ =\left(5x+4\right)\left(x+6\right) \end{gathered}$$

Since the polynomial is factorized, it is not a prime.

Step 2. Factor 4x2+22x+10.

Factorized the polynomial, $4x^2+22x+10$.

$$\begin{gathered} 4x^2+22x+10=4x^2+20x+2x+10 \\ =4x\left(x+5\right)+2\left(x+5\right) \\ =\left(4x+2\right)\left(x+5\right) \end{gathered}$$

Since the polynomial is factorized, it is not a prime.

Step 3. Factor 4x2+38x+70.

Factorized the polynomial, $4x^2+38x+70$.

$$\begin{gathered} 4x^2+38x+70=4x^2+28x+10x+70 \\ =4x\left(x+7\right)+10\left(x+7\right) \\ =\left(4x+10\right)\left(x+7\right) \end{gathered}$$

Since the polynomial is factorized, it is not a prime.

Step 4. Factor 5x2+3x+4.

List the factors of 20 and look for the pair with sum of 3.

Factors of 20	Sum of factors
1	20
2	10
4	5

There are no factors with sum of 3. Hence, the given polynomial cannot be factored using integers. Therefore,$5x^2+3x+4$ is prime. As the obtained answer completely matches with option D, option D is the required answer.