Question: Factor the trinomial completely. \[ 25x^4 + 40x^2 + 7 \] Select the correct choice below and,…

Factor the trinomial completely.

\[ 25x^4 + 40x^2 + 7 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \( 25x^4 + 40x^2 + 7 = \boxed{\phantom{0}} \) (Factor completely.)

B. The polynomial is prime.

Solution

To factor the trinomial completely, we need to check if it can be factored or if it is prime. The expression given is: \[ 25x^4 + 40x^2 + 7 \] 1. Observe the degree of each term: The expression is a trinomial in terms of \(x^2\): \[ 25(x^2)^2 + 40(x^2) + 7 \] 2. Look for any factorization pattern or apply the quadratic formula on \(x^2\): \[ a = 25, \quad b = 40, \quad c = 7 \] 3. Calculate the discriminant to check if it has real roots: \[ \Delta = b^2 - 4ac = 40^2 - 4 \cdot 25 \cdot 7 \] 4. Simplify the discriminant: \[ \Delta = 1600 - 700 = 900 \] 5. Since the discriminant \(\Delta\) is a perfect square, the trinomial can be factored: 6. The roots can be calculated as: \[ x^2 = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-40 \pm 30}{50} \] 7. Calculate the roots: \[ \begin{cases} x^2 = \frac{-40 + 30}{50} = -\frac{1}{5} & \\ x^2 = \frac{-40 - 30}{50} = -\frac{7}{5} & \end{cases} \] Since both solutions for \(x^2\) are negative, there are no real \(x\) for which these roots exist. Therefore, the trinomial is considered prime over the real numbers. So, the polynomial is prime. The correct choice is: B. The polynomial is prime.