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We're on problem 17.

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And it asks us which expression shows the complete

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factorization of 12x squared minus 147?

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Let's see, first of all, can we factor out?

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147 is a strange looking number.

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Let's see if we can factor out something.

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It's definitely not divisible by 12, because

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12 goes into 144.

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Is it divisible by 3?

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1 plus 4 is 5, 5 plus 7 is 12, and 12 is divisible by 3.

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And I don't know if you've seen that before.

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You just add up the digits and if that is divisible by 3,

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then the number's divisible by 3.

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So 147 is the divisible by 3.

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Let's see what it is.

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Let's work it out.

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It's 4 times 3 is 12, 27, it's 49.

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So we can factor 3 out of this.

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So then it becomes 3 times-- what's 12 divided by 3-- 4x

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squared minus-- 147 divided by 3, which we just

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figured out, was 49.

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And now this, once again, looks just like a squared

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minus b squared, where a would be 2x and b would be 9.

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So now we can factor that more.

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And I'll just switch colors arbitrarily.

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That becomes 3 times a plus b, so that would be 2x plus 7

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times a minus b.

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2x minus 7.

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And that is choice D.

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In choice D, they wrote the 2x minus 7 first, but it's the

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same difference.

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So it's choice D.

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Problem 18.

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Let me see if I can get problem 18 going.

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Problem 18.

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Well, maybe I should cut and paste it.

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Why not?

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So I copied it and then let me paste it here.

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That's the problem.

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I'll rewrite it because I don't if that's big enough for

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you to see.

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So it says x plus 3 over x plus 5 plus 6 over x squared

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plus 3x minus 10.

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So when you add fractions, whether you're doing it with

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algebraic fractions or regular fractions, you have to find a

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common denominator.

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And let's see, we have to find the least common multiple of

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the denominators here, but I have a suspicion that this x

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plus 5 goes into this.

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So let's see if I can factor this.

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What two numbers, when I multiply them, equal minus 10,

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and when I add them equal plus 3?

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Let's see.

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Well, 5 times what is minus 10?

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It's 5 times minus 2 and 5 plus minus 2 is 3, so that

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works. x minus 2.

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So this is actually our least common multiple.

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This expression right here.

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So let me write it that way.

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So that is equal to-- I'm just going to rewrite this as x

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plus 5 over x minus 2.

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Now, x plus 3 over x plus 5, if we were to multiply both of

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those times x minus 2 to get something in this form, this x

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plus 3 over x plus 5 is the same thing as x plus 3 times x

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minus 2 over x plus 5 times x minus 2, right?

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You could just cancel this out right here and

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you'd get back to that.

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And now we're adding that.

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We're adding this term to this term.

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6 over this, well, this is the same thing as this, so that's

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just plus 6.

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And now we just get into simplification mode.

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So x times x is x squared.

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x times minus 2 is minus 2x.

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3 times x is plus 3x.

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3 times minus 2 is minus 6, so that's this term right here.

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Got us that.

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And then we have the plus 6.

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And then all of that is over this stuff.

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And I look at the choices and it seems like they have it in

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this form, so I'll just write it in that form.

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x squared plus 3x minus 10.

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And let's see, the minus 6 and the plus 6 cancels out, and

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we're left with minus 2x plus 3x.

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So that's x squared-- minus 2 plus 3-- plus x over x squared

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plus 3x minus 10.

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And that is choice A.

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Choice A.

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Next problem.

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I'm almost out of space.

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I'll draw a line here, just so you don't get distracted.

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What is the simplified form of, and they write 3a squared

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b to the third, c to the minus 2, all of that over--

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interesting-- a to the minus 1 b squared c, and all of that

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is to the third power.

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So let's get in simplification mode.

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So this bottom part, we can re-simplify as-- let's see,

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maybe I wrote too big.

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3a squared, b to the third, c to the minus 2.

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All of that, it's a to the negative 1, b squared, c to

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the third power, that's each of these items

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to the third power.

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So a to the minus 1 to the third power.

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You can multiply the exponents.

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That becomes a to the minus 3.

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I just took the negative 1 times the 3.

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b squared to the third power.

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That's b to the 2 times 3.

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That's b to the sixth power.

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And then finally c.

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Well, that's just c to the first of the third power, so

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that's c to the third.

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And now we can just say, well, this is the same thing.

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Let me switch colors.

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This color's getting mundane.

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This is equal to 3 times a to the 2-- we're dividing by a to

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the negative third-- so it's 2 minus minus 3.

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Let me write that, just so you understand-- 2 minus minus 3

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power-- that's where I got the 2; that's where I got the

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minus 3, and I subtracted because I'm dividing-- times b

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to the 3 minus 6, times c to the minus 2 minus 3.

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Once again, if we were multiplying these two, I would

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add the exponents.

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But anyway, let's simplify this.

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This equals 3a, 2 minus minus, so that's plus, 3a to the

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fifth, b to the minus 3, c to the minus 5.

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And this is the same thing as 3a to the fifth.

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b to the minus 3 is the same thing as 1 over b to the

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third, so over b to the third.

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And this is the same thing as 1 over c to the fifth.

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And that is choice A.

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Choice A.

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Next problem.

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Next problem.

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Oh, we already finished that page.

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All right.

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Let me copy and paste what they wrote.

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Let me put it at the top of this right there and paste it.

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That's what they're asking and I'll write it.

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20x to the minus fourth over 27y squared

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divided by this fraction.

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So first of all, when you divide by a fraction, that's

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the same thing as multiplying by the inverse, right?

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So let's do that.

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I want to get rid of this pesky-looking division sign.

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So let's rewrite this as 20x to the minus fourth over 27y

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squared times-- instead of dividing by this, let's

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multiply by the inverse.

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So what's the inverse?

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15y to the minus 5 over 8x to the minus 3.

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I just flipped it.

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All right?

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So let's see what we can do here.

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It seems like a lot of these numbers have common factors.

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Let's see, if we divide 15 by 3 and 27 by 3, so

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this becomes 5.

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27 divided by 3 becomes 9.

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And just so you can think, you can view this as one

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continuous denominator or one continuous fraction.

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20x to the minus 4 times-- well, now it's 5y to the minus

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fifth divided by 9y squared.

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Because when you multiply fractions, you just multiply

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the numerator times the numerator divided by the

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denominator times the denominator.

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Anyway, let's just keep simplifying.

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If you divide this by 4, you get 5.

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If you divide this by 4, you get 2.

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Let's see, I don't want to do too many steps all at once.

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You get 5x to the minus fourth times 5y to the minus fifth,

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all of that over 9y squared times 2x to the minus 3.

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So let's see.

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Let's get all the numbers out.

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So that is equal to 5 times 5 is 25 over 9 times 2 is 18

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times x-- let's do the x-- x to the minus 4-- minus because

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we're dividing-- minus 3.

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Minus minus 3 times y to the minus 5 minus 2, because we're

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dividing by that one.

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And that is equal to 25/18.

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Let's see, a minus minus, so that becomes a plus.

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Minus 4 plus 3, x to the negative 1, and then minus 5

185
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minus 2, y to the minus 7.

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187
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This is the same thing as 1 over x to the 1.

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This is the same thing is 1 over y to the 7.

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So this is equal to 25 over 18x, 1

190
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over x, y to the seventh.

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y to the seventh in the denominator is the same thing

192
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as y to the minus seventh in the numerator.

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Anyway, that is choice-- let's see, 25 over 18--

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that is choice D.

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Choice D.

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And I'm out of time.

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See you in the next video.

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