Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College solving polynomial equations. If a * b = 0 then a = 0 or b = 0. So our goal is to get zero on one side and then to factor until we have a monomial. So we can set each term equaling to 0. So t ^2 - 3, T -28 = 0. It's currently a trinomial on the left side, and I need it to be a monomial. So -28. What two things multiply to give me -28 but add to give me -3? So we're going to see that it's going to be 4 and seven, with it being a -7. So t + 4 and t - 7. So now what we do is we actually set t + 4 equal to 0 because it's a monomial, and t - 7 equaling 0. So we can see T equal -4 and t = 7. Frequently we write the solution set when it's just a listing of numbers with braces and the actual numbers inside. This next example we need to get everything to one side again and this one is going to be a perfect square. So we have r * r and -4 and -4 and that equals 0. So r - 4 = 0 and r - 4 = 0 R equal 4. In both cases, when we write our solution, we only need to list for one time, and this is actually called a double root or a double solution because the same number occurred more than once. In our next example, we're going to have The X Factor out and get X -, 9 equaling 0. If we factor out a variable in the front IEA letter, that variable portion is going to equal 0. So we have X equaling 0 here, X equaling 9 here. So our solution is 0 and 9. An important piece that we could think about also is the highest exponent tells us how many solutions to expect. So I expected 2 solutions, 0 and 9. Back here in #2 I expected 2 even though the two were the exact same. So in #4 we're going to expect 3 solutions, the highest exponent. So we're going to pull everything to one side so that zero is on the other. Then we're going to factor out anything they have in common, in this case an X. We're going to look for two numbers that multiply to give me -63 but add to give me -2. So 63 one and 63 three and 21 seven and 9:00. So that's going to be our number and we want +7 and -9 because they needed to add to give me a -2. So now once we have this all the way to a monomial, just things that are being multiplied and divided. Remember, the parentheses act like a single item. We set each piece equal to 0 and there are three pieces in this one. So we have X = 0, X equal -7, and X equal 9. When we list these, we usually list them smallest to largest, so -7 zero and 9:00. The next example T to the 4th. So we're going to have t ^2 and t ^2. We need 2 numbers that are going to multiply to give me 36 but add to give me -13. So 1 and 36, two and 18, three and 12/4 and 9/6 and six. If I need them to add to give me a -13, we're going to use -4 and -9 right here. Negative times. A negative made the positive. Well, this is still 2 difference of squares, so we get t + 2 T -2 T plus three, t -, 3, and we should expect 4 answers because the highest exponent in the original. So we're going to get T equal -2, T equal, 2T equal -3 and three and set notation -3 negative 223. We usually list lowest to highest, or if we were thinking about on a number line left to right, this next one doesn't equal 0, but it's already factored. The fact is that that doesn't do us any good. We're going to actually have to redistribute, multiply it out, foil, whatever, so that we can get zero on one side. Once we get the zero on one side, then we need to refactor and set each of those terms equal to 0. So we're going to get A = 6 and A equal -6 our set notation -6 to 6. The next example, we want to find A when F of A is five. Well, F of A is sticking A in every time we see the unknown that was in the parentheses. So first of all, just plain F of A is a ^2 + 14, A+ 50. But we're told that F of A is going to equal 5. If F of A equal 5 and F of a equal this equation, then the two things have to equal each other by the transitive property. So a ^2 + 14 A+ 50 equal 5. Now it turns into one that we've already done. We're going to get everything to one side and turn it into a monomial SO2 numbers that are going to multiply to give me 45 one and 45 3 and 15 five and 9:00. But add to give me 14, so 5 and 9. So a + 5, A plus 9. So a is going to be -5 or -9. So the two possibilities for a -9 negative 5. This last one, let's take everything to one side. 6B squared -13 B +6. It doesn't have a one coefficient, so we're going to use some of our factoring strategies. We need 2 numbers that are going to multiply to give me 36. And there's a lot of possibilities here. Well, maybe not as many as I thought. And I need the ones that's going to add to give me -13 so -4 negative 9. So b - 4 / 6 and b - 9 / 6. Simplify this so we get b - 2/3 and b - 3 halves. So three b - 2 and two b - 3 is our factorization made, making it a monomial. Now we're going to take each of those and set them equal to 0. When I do this, I'm going to add 2 and then divide by three. I'm going to add 3 and then divide by 2. So my solutions are 2/3 and three halves. Now here, it's actually kind of cool if we looked right back here at this piece and set each of them equal to 0. There we can see there is my 2/3 solution and there's my three halves solution. We have many for you to try on the next page. Thank you and have a wonderful day.