Hello wonderful mathematics people. I'm Anna Cox from Kellogg Community College. Factoring trinomials with a leading coefficient. The first thing we're going to do is we're going to make sure that there's no common factors. If there is, we need to factor it out of all three terms. Then we're going to look at the leading coefficient and multiply it by the constant. So in this case, 6 * -35 we're going to list all pairs that multiply to give us -210 So 1 and 210, two and one, O 5, three and 75 and 42, six and 35, seven and 30-10 and 21-14 and 15. Once we do that, we need it to be a -210. So we know that one number is going to be positive when the other is negative. If we look at this very first example, I want them to add to give me a +1. If I want them to add to give me a positive, I know that the bigger number needs to be the positive number, so the smaller one would be the negative. If we look at the sums here, we'd get 209-103, 67-37, 29-23-11, and 1. So we need this last pair here at the bottom. So one method to factoring would be to write X - 14 over this leading coefficient of 6 and X + 15 over the leading coefficient of 6. Reduce those fractions so 14 six turns into 7 thirds. 15-6 A3 comes out of each of those, and that turns into five halves. Now take the denominator and put it in front of the coefficient. So three X -, 7 * 2 X plus 5 is what six X ^2 + X -35 factors into. We can always check by foiling. 3X plus 2X is 6 X squared. Three X * 5 is 15 X -7 * 2 X is -14 X and positive 15X and -14 X does give me the One X -7 * 5 is -35. Let's look at the next example. We need it to be a +11, so we're going to have this -10 and +21. So if I do X -, 10 over the leading coefficient of six, X + 21 over the leading coefficient of 6, I need to reduce those. So X -, 5 halves, X + 7 halves. If there's a denominator, we're going to move it in front of the leading coefficient. And 10-6 is not 5 halves, it's 5 thirds. So three X -, 5. That's hard to read. Let me erase that and write it again. Three X - 5 * 2 X +7 3X times 2X is 6 X squared. Three X * 7 is 21 X -5 * 2 X is -10 X which combines to give a negative +11 X -5 * 7 gives the -35. Let's look at this next example. I need it to be a -29 now. Well, here's my 29. How do I make it a negative? I'm going to change the sign so that it's going to be a +6 and a -35. So if we do X + 6 over the leading coefficient of 6 * X -, 35 over the leading coefficient of six, 6 / 6 is 135 / 6 doesn't reduce. So we're going to take that six and put it in front of the X. And that's our factored form. We can double check it by multiplying it out. Let's look at a few more examples. Here I changed the coefficients to be 8 and 15. The first thing, do they all three have anything in common? And the answer's no. So I'm going to multiply the 8 and the -15 to get -120. So this time -120 I need one to be +1 to be negative. If I'm looking to make it a -2, I'm going to want the bigger one to be a negative this time. So then when I look and add these, I'd get -119 negative 58, negative 37, negative 26, negative 19 -14, negative seven -2 These are actually all the possibilities for the middle number. If the middle number is anything other than one of these or the positive of these, then it wouldn't factor. So we want this -2 So we're going to use the 10 and the negative twelve X + 10 over the leading coefficient, X -, 12 over the leading coefficient. Reduce those 10 eighths is 5 fourths, 12 eighths. A four comes out of top and bottom, so 3 halves. If it's got a fraction, put that denominator in front of the variable. So four X + 5 two X - 3 does indeed factor into the eight X ^2 - 2 X -15. If we look at this next example, we want the middle term to be a -14. So here's the six and the negative twenty X + 6 over the leading coefficient of eight X -, 20 over the leading coefficient of 8. So X + 3/4 X minus A4 comes out of both, so 5 halves, putting the leading coefficient front four X + 3 to X -, 5. If we check it in foil, this does indeed work. This next one a negative 37. So we have 3 and -40 X plus three over the leading coefficient of eight, X -, 40 over the leading coefficient of eight. So 3/8 doesn't reduce. So we're going to put that 8IN front and get eight X + 340 / 8 does reduce to five. And that's our factored form. Just a couple more examples. If we use the same coefficients but now the middle term is a -43 and it's a +120 when we multiply the leading coefficient, the constant, we would add these together to find the pair that would give us the middle number. In this case we need the middle number to be a negative. So a negative times a negative would be a positive, negative times negative, negative times negative, negative times negative. So all these sums would be negative and we would look for 43 negative 43. So we want -3 and -40 X -3 / 8 X -40 / 8 three eights doesn't reduce, so the eight goes in front. The 40 / 8 does reduce to a positive whole number of or a negative whole number of five. Here we want it to be negative 23. So we're going to have X - 8 / 8 * X -, 15 / 8. Eight over 8 reduces to 115 / 8 doesn't reduce. So the eights going to go in front of that coefficient. The last example it's a positive 26, so I need it to be a positive number times a positive number to add to give me a positive. So if I set X + 6 divided by the leading coefficient of eight, X + 20 divided by the leading coefficient of eight, we'd reduce these X + 3/4 X plus four goes into both of these. So 5 halves we're going to take and put that denominator in front of the variable. We can always double check by foiling it out. Thank you and have a wonderful day. This is Anna Cox.