factor following

Question

Factor following polynomials.

                              x² + 2x -15

Solution

x² +2x -15

Okay since the first term is x² we know that the factoring has to take the form.

x² + 2 x -15 = ( x + _  )( x + _  )

We know that it will take this form as while we multiply the two linear terms the first term has to be x² and the only way to get that is to multiply x by x. thus, the first term in each factor has to be an x. To finish this we just have to determine the two numbers that has to go in the blank spots.

We can narrow down the possibilities significantly.  Upon multiplying the two factors out these two numbers will have to multiply out to get -15.  In other terms these two numbers has to be factors of -15.  Here are all of the possible ways to factor -15 via only integers.

                       ( -1)(15)           (1) (-15)          ( -3) (5)           (3) ( -5)

Now, we can only plug these in one after another & multiply out till we get the exact pair. Though, there is another trick which we can use here to help us out. The accurate pair of numbers has to add to get the coefficient of the x term.  Thus, in this case the third pair of factors will add to "+2" & so that is the pair we are after.

Following is the factored form of the polynomial.

                                            x² + 2 x -15 = ( x - 3) ( x + 5)

Again, we can always verify that we got the right answer my doing a quick multiplication.

Note that the method we utilized here will only work if the coefficient of the x² term is one.  If it is anything else it won't work and we actually will be back to trial & error to get the accurate factoring form.