Bottoms up Factoring Method

Welcome to the Student Academic Learning Services video on the Bottoms Up factoring Method for trinomials. In this video we are simply going to go through an example using the bottoms up factoring method. Our example is 6 x squared plus 13 x minus 5. The very first thing we are going to do is see if we can common factor anything out of this particular trinomial. As there is nothing that goes into 6, 13 or negative 5 and they don’t share any common variable they don’t have a common factor so we are going to start factoring this as is. The next step in the bottoms up factoring method is to just draw an x on the page you can see we already have one in the middle and the first thing we are going to do with this is we are going to take the coefficient of our middle term so that 13 we are going to take that and place it in the top section of our x. The next step will be to take the product of our first and last terms. So 6 times negative 5 and we are going to place that in the bottom section of our x. So 6 times negative 5 gives us negative 30. Our next step in the method is to figure out two numbers that multiply to negative 30 and add to positive 13. So if we look at all the numbers that multiply to 30, we see that 2 and 15 are the numbers we are going to look at here. So if we have positive 15 and negative 2 they will multiply to negative 30 and add to positive 13 so that’s what we are going to use. We are going to take one of those numbers and place it in the left hand section and take the other number in the right hand section. We could have flipped these and placed the negative 2 in the left hand side and the positive 15 in the right hand side, it makes no difference. Our next step is to take our leading coefficient so the 6 and we are going to place that under the 15 and negative 2, essentially creating two fractions. Once we have our two fractions we now want to convert them to lowest terms. So on the left hand side we have 15 over 6, dividing both of those by 3 we get 5 over 2. And on the right hand side we have negative 2 over 6, dividing both of them by 2 will give us negative 1 over 3. Now we go “bottoms up” this is where the method gets its name. And the 2 in the left hand section on the denominator is going to be our leading term in our factored bracket. So the 2 becomes the first term and we just attach an x to it and our 5 becomes our second term. And on the right hand side our 3 the denominator is going to become our leading term in the bracket and the negative 1 becomes the second term. So we get 3 x minus 1. Very important to remember the signs here, this wouldn’t be correct if we had said positive 1 in that second bracket. Now that we have our factors we want to check to make sure they are right. We do that by expanding our factors out and if we get back to our original trinomial we know we are correct. So using FOIL we have 2 x times 3 x gives us 6 x squared. 2 x times negative 1 gives us negative 2 x. Positive 5 times 3 x gives us positive 15 x. And lastly, positive 5 times negative 1 gives us negative 5. Combining the like terms we come up with 6 x squared plus 13 x minus 5. This is our original trinomial so we know the factors of 2 x plus 5 and 3 x minus 1 are the correct factors. This concludes our video on the bottoms up method for factoring trinomials. If you have any questions about the material covered in this video I strongly encourage you to come to by the SALS Centre and make an appointment with the appropriate Learning Skills Advisor. Thanks for watching.