Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College solving polynomial equations. If a * b = 0, then we know that either a = 0 or b = 0. We also will know based on the degree of the polynomial how many solutions there should be. So if it's a second degree equation, we should expect 2 solutions. A third degree equation we should expect 3 solutions. 4th degree equation we should expect 4 solutions. 50th degree equation we'd expect 50 solutions. So the first thing on our examples is we need to always make sure it equals 0. In this first example, it does not equal 0, so we're going to subtract that 28 so that we do have one side equaling 0. Now it's not multiplication, it's not a monomial on the left side. So we need to factor t ^2 -, 3 T -28. The two numbers that multiply to give us -28 and add to give us -3 are going to be -7 and +4. So now we're going to set each of those equal to zero. T - 7 = 0 and t + 4 = 0. So T is going to be 7 or T is going to be -4. The reason we have two answers is it was a second degree equation, so we expected 2 answers. This next example, we're going to start by getting everything to one side so that we have 0 on the other. It's not a monomial, so we need to factor it. What times what gives us 16 but adds to give us -8 and it's going to be -4 and -4. This case, we're going to set each of those parentheses equal to 0, so we get R equal 4 and R equal 4. We actually have two solutions here, even though they're the same solution. It's called a double root or a double solution. So R equal 4 occurred twice, but it really was 2 solutions. This next one, we're going to start by factoring out an X. If we have an X factored out, we now know that that X has to equal 0 and the X -, 9 has to equal 0. So we know that X is going to be 0 or 9 this next one. We need to get everything to one side so that we have 0 on one side. Then we need to ask if it's a monomial, and in this case it's not. So we're going to factor out anything they have in common 1st and they have an X in common. Then we want to think about what 2 numbers multiply to give us -63 but add to give us -2 and our answer is going to be -9 and +7. Now we have 3 things that are being multiplied together to give us the monomial, and this was a cubic equation. So we should expect 3 answers. We're going to get zero negative 7:00 and 9:00. We usually write them smallest to largest, so -7 zero, then nine. This next one, how many should we get? We should get 4 because of the 4th degree. What 2 numbers multiply to give us 36, but add to give us -13? And the answer is going to be -9 and -4 multiply to give us +36 and add to give us -13. But now t ^2 - 9 is really the difference of squares, so t - 3 and T plus three t ^2 - 4 is also the difference of squares t - 2 and t + 2. So now we're going to have four different parentheses, each equaling 0, and then we're going to solve them. So that first parenthesis we're going to get T equal 3, and then T equal -3 T equal, 2T equal -2. We list these the lowest to the highest. If you thought about reading on a number line, it's left to right, so -3 negative 2, two, and three. Well, what if it doesn't equal to 0 to start? That's OK, but we actually have to get it equal to 0. So we're going to multiply things out so that we get everything on one side with zero on the other. So we foiled the a -, 4 A+ 4, which is really different to squares, and we brought the 20 over. When we combine our like terms, we get a ^2 - 36 = 0. Well, a ^2 - 36 is the difference of two squares A - 6, A+ 6, so a - 6 = 0, A+ 6 = 0 so a = 6 and A equal -6. This next one we're given F of X = X ^2 + 14 X plus 50, and we want to find A when F of a equal 5. So we're going to stick A in every time we see our X. If we stick A in, every time we see our X, we're going to get F of A equaling a ^2 + 14, A+ 50. But F of A is given as equaling the number 5. So we're going to actually take and put five in instead of that F of A. Now, is one side equaling 0? At the moment, no, so we're going to subtract the five. I like my leading coefficient to always be positive, so I'm going to subtract the five versus the a ^2 -, 4 + 14 A and plus 52 numbers now that are going to multiply to give us 45, but add to give us 14, nine, and five. And we're going to set each of those parentheses now that it's a monomial equal to 0, so A equal -9 and A equal 5. Let's look at one more. What if we had some leading coefficients? What if we had 6B squared +6 equaling 13B? Well, we would start by having to take everything to one side and now we need to factor that. Well, one way to factor it is to look at the leading coefficient and the constant. What numbers are going to multiply to give me 36 but add to give me -13? And if we list all of our possibilities, we're going to realize it's going to be the four and the 9. So if we went b -, 4 sixths and b -, 9 sixths, we're going to reduce those fractions b -, 2/3 and b -, 3 halves. Now take that denominator out in front. This is just an alternative method to factoring. Now we have this factored so that when we foil it out, we get six b ^2 -13 B and +6 equaling 0. So we're going to set each of these parentheses equal to 0 and we would get 3B equal 2 or B equal 2 thirds, 2B equal 3 or B equal 3 halves. When we list them, we go smallest to largest. Thank you and have a wonderful day.