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

B. The trinomial is prime.

Solution

To factor the trinomial \(26x^2 + 63x - 98\), we will use the method of factorization by grouping. First, look for two numbers that multiply to \(26 \times (-98) = -2548\) and add to \(63\). These numbers are \(98\) and \(-35\). Next, rewrite the middle term \(63x\) using these numbers: \[ 26x^2 + 98x - 35x - 98 \] Now, group the terms: \[ (26x^2 + 98x) + (-35x - 98) \] Factor out the greatest common factor from each group: \[ 2x(13x + 49) - 7(5x + 14) \] Notice that we should actually have: \[ 2x(13x + 49) - 7(5x + 14) \] On regrouping correctly, the expression should factor as: \[ (2x - 7)(13x + 14) \] Thus, the trinomial \(26x^2 + 63x - 98 = (2x - 7)(13x + 14)\).