Problem 71 - Introductory Algebra for College Students

Question

Factor by grouping. $$x^{2}+3 x-5 x-15$$

Step-by-Step Solution

Answer

The factorization of the polynomial \(x^{2}+3x-5x-15\) by grouping is \(x(x - 2) - 15\).

Step 1: Combine Like Terms

First, combine the terms in the polynomial that are similar, to simplify the equation. The polynomial \(x^{2}+3x-5x-15\) simplifies to \(x^{2} - 2x - 15\).

Step 2: Group the Terms

The next step is to group the terms that have common factors. In this case, it's best to group the \(x^{2}-2x\) together and \(-15\) separately. The grouped polynomial then becomes \((x^{2}-2x) - 15\).

Step 3: Factor the Groups

Factor out the common factor from each group. From the first group, \(x\) can be factored out as a common factor, leaving \(x(x-2)\). The second group is already as simplified as possible. So, the polynomial is now \(x(x - 2) - 15\).

Step 4: Final Expression

Notice that the polynomial now is in the factored form, and we cannot further simplify it. So the final expression is \(x(x - 2) - 15\).