Question: Factor the trinomial. \[ 26x^2 + 63x - 98 \] Select the correct choice below and, if necessary,…

Factor the trinomial.

\[ 26x^2 + 63x - 98 \]

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

A. \( 26x^2 + 63x - 98 = \boxed{\phantom{0}} \) (Factor completely.)

B. The trinomial is prime.

Solution

To factor the trinomial \(26x^2 + 63x - 98\), follow these steps: First, look for common factors in the coefficients 26, 63, and -98. There is no common factor, so we proceed with factoring by grouping or the ac method. Identify \(a = 26\), \(b = 63\), and \(c = -98\). Calculate \(a \cdot c = 26 \times -98 = -2548\). Look for two numbers that multiply to \(-2548\) and add to \(63\). The numbers are \(127\) and \(-20\). Rewrite the middle term \(63x\) using these two numbers: \[ 26x^2 + 127x - 20x - 98 \] Group terms: \[ (26x^2 + 127x) + (-20x - 98) \] Factor out the greatest common factor from each group: \[ x(26x + 127) - 2(10x + 49) \] Recognize the factors: \[ x(26x + 127) - 2(10x + 49) \] Since we cannot find simple factors for each binomial group, check if the trinomial might actually be prime. After checking again, we find: - \(26x^2 + 63x - 98\) cannot be factored further with simple integer factors. - Therefore, the trinomial is likely prime. Select option B: The trinomial is prime.