Learn about CA Algebra I: Factoring Quadratics Video

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Learn about CA Algebra I: Factoring Quadratics

Learn about CA Algebra I: Factoring Quadratics All right we’re on problem 44. They said, which is the factored form of 3A2– 24AB + 48B2. And if this confuses you with A’s and B’s instead of an X there I just like to think of, in this case this looks like an X2 so I like to think of it the same I would like to think of it in terms of it this was X2 minus some numbered times X plus some other numbers. So let’s put that in the back of your head, if that confuses you don’t worry about it but the first thing I want to do is try to get rid of the coefficient on the in this case the A2 term. And let’s see if we can factor out a number, well sure all of this is divisible by three, right, three is divisible by three, 24, is divisible by three and 48 is divisible by three. So we should be go to factor out of three so let’s do that. So that is equal to 3 x A2– 8AB, 24 divided by three is eight, and there’s a minus in front of it plus 48 divided by 16 is three, 48 divided by three is 16B2. Now to factor this we have to think about it as are there two numbers we’ll think of it let me rewrite this a little bit. If we view A is kind of the independent, you wouldn’t have to do it but I just wanted you to visualize it properly 3A2– 8B(A+ 16B2) so if we viewed A2 is kind of the independent variable or the X term, so now this kind of has a shape of polynomials and hopefully you are used to factoring a little bit, we just have to think, are there two numbers that add up to -8B and that when I multiply the two numbers I get 16B2 right, so first of all the number is going to be in terms of B obviously because if I’m adding I’m going to get a B and I get B so if I square them, I might get a B2 and then also makes me lead to the as the same number just used twice otherwise it would be weird to get squared here. So the number that should pop up is okay what two numbers add up to –8 and then when it square it is equal to 16– 4 and -4, -4 is -8, -42 is 16.So the number we’re talking about is -4B+ -4B = -8B and -4B2 = 16B2. So we can factor this out to be 3— and let me switch colors times we said -4B is a number when you add twice we get -8b and you squared to get that so that’s A– 4B( A-4B) and that is choice C. Next problem now let’s get 45. Let me copy and paste 45 in here let’s get problems 45 and copy and past it in. Over do this on top of this one and see if we can erase this and look at and now I can paste 45 and ready to go. Which is a factor of X2– 11X+ 24. So there’s two ways to do it. You could just factor this and that, well that’s the easiest way or you could test for zeros but we’ll just factor. So if we wanted to say, well let me just X2– 11X+ 24. So just like the last problem you have to say, what two numbers when add them is equal to -11 and when I multiply them is equal to positive 24? So first of all if we multiply them and when I get a positive 24 they’re either both positive or they’re both a negative. Since if this is a negative number it tells me they’re both can’t be positive, you can’t add two positive numbers and add and end up with a negative. So we’re going to deal with two negative numbers so it tells you what two negative numbers when they add them equals -11 and when I multiply them it is equals 24 and hopefully 8 and 3 pop out that is -8+ -3 = -11 and then -8 times-3 is +24. So this can be factored as X– 8 times X– 3. And then if we look at the choices they had X– 3 there. And of course don’t get confused if they asked you if this was had solved this equation right, this is just an expression now it becomes an equation if we put equal sign. Then we would have this equaling zero and then the—you could say the roots of this polynomial, the solutions of the equation and what makes this true and then the solutions wouldn’t be -8 and -3. The solutions would what makes it zero so the X = 8 would make this zero, or X = 3 will make that— but anyway, I’m going of on a tangent that’s now what they ask you. I think that confuses people sometimes. Okay 46, which of the following shows 9T2+ 12T+ 4 factored completely. Now this is an interesting one because immediately when I look at the numbers there’s not one number that I can just factor out of everything, it’s not like 9,12, and 4 they don’t have any common factors. So I can’t just do that simplification, so we’re going to have to do a little bit more complexity but the best way to think about it is, whatever is on the T2, this is probably kind of this whole expression, if we’re trying to factor it into two binomials or into one binomial, this is going to be the first term of that binomial squared. So we’re going to be dealing with something like 3T right, I’m just using the square root of 9T2 3T plus some number, let’s say plus A times 3T+ B. And now we can just actually multiply this out and see what happens and it will first of all, they gave us a multiple choice. We could just multiply this out and see what happens but let’s pretend like they didn’t give us choices and we have to do this in a vacuum, if we do this in a vacuum we would have to factor ourselves we wouldn’t just be able to test our choices so let’s do that. So if we multiply this out we have 3T times 3T is 9T2 then you have 3T times B, so it’s plus 3BT+ 3AT+AB. So this simplifies to 9T2 + .Now what are adding? You see it says 3 times A+ BT. So it would be 3 times A+ BT and we have to add this two terms and I factored out the T and the 3+AB. Well now we can do a little pattern matching right we could say A+ B times 3 = 12. So A+ B = 12, right this whole coefficient right here is a 12 up here, So A + B = 4 because 4 times 3 = 12, A + B = 4 and we have A times B = 4. So the only number that I can think of, when I add them I get 4 when I multiply them I get 4 is 2 and 2. Right both of these are 2. So, if I were to factor this completely I get 3T+ 22 essentially because they both of this terms of the same thing and that’s choice A. A faster way frankly if you want to do this I might have just been to multiply and I said that is the same thing. Anyway next problem, what is the complete factorization of 32– 8Z2 so let’s think about this a little bit. So the first thing that I like to do when I get this is to try to factor out any numbers that are just common to all of the terms, so let me do that. So 32– 8Z2, 8 goes in the both of these, so let’s factor out an 8 and actually let’s factor a -8 and I’ll show you why I did like because I like to put the Z2 like that and would be positive. So let’s factor out a negative 8 and so you get a -8 and you didn’t have to do that you can just factor an 8 but what’s 32 divided by -8? That’s minus 4, right. And then -8Z2 divided by -8 is just positive Z2 and so we can rearrange this so this becomes a -8 times Z2– 4. And now this is—I’ll review this because this is an Algebra 1 test so this maybe this isn’t obvious to you but in general if you see something like this A + B times A– B, and this might be good exercise for you to multiply this out but this is equal to A2– B2 because the middle terms cancel out and that’s a good exercise for you to do. So this has that same property right this is A2– B2, this is a perfect square. So this up here can be factored as -8 times 8 this is A2– B2 so this is going to be A—so if we could say A is Z and B is 2 right because that may be a Z2– 22 which is 4. So that’s if A is Z, so its Z+ 2 times Z– 2. This is what the most common thing you’ll see in a lot of factoring examples and you’ll see something that has it’s pattern A2– B2 and you should really be able to recognize it and you should be able to prove it to yourself as well that this is the same thing as A+ B times A– B or in this case Z+ 2 times Z– 2 and that is Z– 2. So that’s interesting and this is something. So I don’t see exactly what we wrote. I have a minus -8 but we want to Z+ 2 and Z– 2 but they don’t have that. What they do have is this let me rewrite it, they don’t have any of it but maybe they have they want the minus to be multiplied by one of these other terms. So what we could do is we could multiply the minus because you know multiplication is associative, it doesn’t matter what order you do with in so we could rewrite this as 8 times Z+ 2 times -1 times Z– 2 and then this becomes 8 times Z+ 2. This isn’t minus 1 this is times -1 times what minus Z+ 2 and now I think that choice is there. So we can rewrite this as 8 times 2+ Z—I’m just rearranging it—times 2– Z and that is choice B. And so really given the choices they gave us, it probably would have been faster to just factor out an 8 in front of the beginning and not a positive 8 and you immediately got into 2+ Z times 2– Z. Anyway I’ll see you in the next video.

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Learn about CA Algebra I: Factoring Quadratics

All right we’re on problem 44. They said, which is the factored form of 3A2– 24AB + 48B2. And if this confuses you with A’s and B’s instead of an X there I just like to think of, in this case this looks like an X2 so I like to think of it the same I would like to think of it in terms of it this was X2 minus some numbered times X plus some other numbers. So let’s put that in the back of your head, if that confuses you don’t worry about it but the first thing I want to do is try to get rid of the coefficient on the in this case the A2 term.

And let’s see if we can factor out a number, well sure all of this is divisible by three, right, three is divisible by three, 24, is divisible by three and 48 is divisible by three. So we should be go to factor out of three so let’s do that. So that is equal to 3 x A2– 8AB, 24 divided by three is eight, and there’s a minus in front of it plus 48 divided by 16 is three, 48 divided by three is 16B2. Now to factor this we have to think about it as are there two numbers we’ll think of it let me rewrite this a little bit.

If we view A is kind of the independent, you wouldn’t have to do it but I just wanted you to visualize it properly 3A2– 8B(A+ 16B2) so if we viewed A2 is kind of the independent variable or the X term, so now this kind of has a shape of polynomials and hopefully you are used to factoring a little bit, we just have to think, are there two numbers that add up to -8B and that when I multiply the two numbers I get 16B2 right, so first of all the number is going to be in terms of B obviously because if I’m adding I’m going to get a B and I get B so if I square them, I might get a B2 and then also makes me lead to the as the same number just used twice otherwise it would be weird to get squared here.

So the number that should pop up is okay what two numbers add up to –8 and then when it square it is equal to 16– 4 and -4, -4 is -8, -42 is 16.So the number we’re talking about is -4B+ -4B = -8B and -4B2 = 16B2. So we can factor this out to be 3— and let me switch colors times we said -4B is a number when you add twice we get -8b and you squared to get that so that’s A– 4B( A-4B) and that is choice C.

Next problem now let’s get 45. Let me copy and paste 45 in here let’s get problems 45 and copy and past it in. Over do this on top of this one and see if we can erase this and look at and now I can paste 45 and ready to go. Which is a factor of X2– 11X+ 24. So there’s two ways to do it. You could just factor this and that, well that’s the easiest way or you could test for zeros but we’ll just factor. So if we wanted to say, well let me just X2– 11X+ 24. So just like the last problem you have to say, what two numbers when add them is equal to -11 and when I multiply them is equal to positive 24?

So first of all if we multiply them and when I get a positive 24 they’re either both positive or they’re both a negative. Since if this is a negative number it tells me they’re both can’t be positive, you can’t add two positive numbers and add and end up with a negative. So we’re going to deal with two negative numbers so it tells you what two negative numbers when they add them equals -11 and when I multiply them it is equals 24 and hopefully 8 and 3 pop out that is -8+ -3 = -11 and then -8 times-3 is +24.

So this can be factored as X– 8 times X– 3. And then if we look at the choices they had X– 3 there. And of course don’t get confused if they asked you if this was had solved this equation right, this is just an expression now it becomes an equation if we put equal sign. Then we would have this equaling zero and then the—you could say the roots of this polynomial, the solutions of the equation and what makes this true and then the solutions wouldn’t be -8 and -3. The solutions would what makes it zero so the X = 8 would make this zero, or X = 3 will make that— but anyway, I’m going of on a tangent that’s now what they ask you. I think that confuses people sometimes.

Okay 46, which of the following shows 9T2+ 12T+ 4 factored completely. Now this is an interesting one because immediately when I look at the numbers there’s not one number that I can just factor out of everything, it’s not like 9,12, and 4 they don’t have any common factors. So I can’t just do that simplification, so we’re going to have to do a little bit more complexity but the best way to think about it is, whatever is on the T2, this is probably kind of this whole expression, if we’re trying to factor it into two binomials or into one binomial, this is going to be the first term of that binomial squared.

So we’re going to be dealing with something like 3T right, I’m just using the square root of 9T2 3T plus some number, let’s say plus A times 3T+ B. And now we can just actually multiply this out and see what happens and it will first of all, they gave us a multiple choice. We could just multiply this out and see what happens but let’s pretend like they didn’t give us choices and we have to do this in a vacuum, if we do this in a vacuum we would have to factor ourselves we wouldn’t just be able to test our choices so let’s do that.

So if we multiply this out we have 3T times 3T is 9T2 then you have 3T times B, so it’s plus 3BT+ 3AT+AB. So this simplifies to 9T2 + .Now what are adding? You see it says 3 times A+ BT. So it would be 3 times A+ BT and we have to add this two terms and I factored out the T and the 3+AB. Well now we can do a little pattern matching right we could say A+ B times 3 = 12. So A+ B = 12, right this whole coefficient right here is a 12 up here, So A + B = 4 because 4 times 3 = 12, A + B = 4 and we have A times B = 4.

So the only number that I can think of, when I add them I get 4 when I multiply them I get 4 is 2 and 2. Right both of these are 2. So, if I were to factor this completely I get 3T+ 22 essentially because they both of this terms of the same thing and that’s choice A. A faster way frankly if you want to do this I might have just been to multiply and I said that is the same thing.

Anyway next problem, what is the complete factorization of 32– 8Z2 so let’s think about this a little bit. So the first thing that I like to do when I get this is to try to factor out any numbers that are just common to all of the terms, so let me do that. So 32– 8Z2, 8 goes in the both of these, so let’s factor out an 8 and actually let’s factor a -8 and I’ll show you why I did like because I like to put the Z2 like that and would be positive. So let’s factor out a negative 8 and so you get a -8 and you didn’t have to do that you can just factor an 8 but what’s 32 divided by -8? That’s minus 4, right.

And then -8Z2 divided by -8 is just positive Z2 and so we can rearrange this so this becomes a -8 times Z2– 4. And now this is—I’ll review this because this is an Algebra 1 test so this maybe this isn’t obvious to you but in general if you see something like this A + B times A– B, and this might be good exercise for you to multiply this out but this is equal to A2– B2 because the middle terms cancel out and that’s a good exercise for you to do.

So this has that same property right this is A2– B2, this is a perfect square. So this up here can be factored as -8 times 8 this is A2– B2 so this is going to be A—so if we could say A is Z and B is 2 right because that may be a Z2– 22 which is 4. So that’s if A is Z, so its Z+ 2 times Z– 2. This is what the most common thing you’ll see in a lot of factoring examples and you’ll see something that has it’s pattern A2– B2 and you should really be able to recognize it and you should be able to prove it to yourself as well that this is the same thing as A+ B times A– B or in this case Z+ 2 times Z– 2 and that is Z– 2. So that’s interesting and this is something.

So I don’t see exactly what we wrote. I have a minus -8 but we want to Z+ 2 and Z– 2 but they don’t have that. What they do have is this let me rewrite it, they don’t have any of it but maybe they have they want the minus to be multiplied by one of these other terms.

So what we could do is we could multiply the minus because you know multiplication is associative, it doesn’t matter what order you do with in so we could rewrite this as 8 times Z+ 2 times -1 times Z– 2 and then this becomes 8 times Z+ 2. This isn’t minus 1 this is times -1 times what minus Z+ 2 and now I think that choice is there. So we can rewrite this as 8 times 2+ Z—I’m just rearranging it—times 2– Z and that is choice B.

And so really given the choices they gave us, it probably would have been faster to just factor out an 8 in front of the beginning and not a positive 8 and you immediately got into 2+ Z times 2– Z. Anyway I’ll see you in the next video.

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