Algebra I
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CA Algebra I: Number Properties and Absolute Value
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Simple Logical Arguments
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CA Algebra I: Graphing Inequalities
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CA Algebra I: Slope and Y-intercept
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CA Algebra I: Systems of Inequalities
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Factoring Quadratics
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CA Algebra I: Completing the Square
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CA Algebra I: Quadratic Equation
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CA Algebra I: Quadratic Roots
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CA Algebra I: Rational Expressions 1
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CA Algebra I: Rational Expressions 2
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CA Algebra I: Word Problems
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CA Algebra I: More Word Problems
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CA Algebra I: Functions
CA Algebra I: Rational Expressions 2 66-70, more simplifying of rational expressions
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- We're on problem 66.
- And it says what is x squared minus 4x plus 4, divided by x
- squared minus 3x plus 2, reduced to lowest terms?
- So they probably want us to factor each of these
- quadratics and see if any of these terms cancel out.
- So let's try to do that.
- So the numerator, this seems pretty easy to factor.
- What two numbers when I multiply them equal 4?
- And when I add them equal minus 4?
- Well it's minus 2, right?
- Minus 2 and minus 2 is minus 4.
- Minus 2 squared is plus 4.
- So this is x minus 2 times x minus 2.
- And you could test it if you don't believe it.
- Multiply that out.
- Divided by, let's see, what two numbers?
- This looks factorable.
- They both have to be the same sign because when you multiply
- them you get a positive.
- And they're both going to be negative, because when you add
- them you get a negative 3.
- So let's see, minus 2 and minus 1.
- Minus 2 times minus 1 is positive 2.
- Minus 2 plus minus 1 is minus 3.
- So x minus 2, times x minus 1.
- And if we assume that x is never equal to 2, because that
- would make this expression undefined, we cancel that out.
- You'll learn later that would cause a hole in the graph,
- because the function is undefined there.
- And you're left with minus 2 over minus 1.
- And that is choice A.
- Problem 67.
- This is good practice.
- They give a bunch of it.
- They say what is-- I'll just write it-- 12a cubed minus 20a
- squared over 16a squared plus 8a.
- Reduce to lowest terms. So let's just try to factor out
- things on the top and the bottom and see what happens.
- So in the top, in the numerator-- let me switch
- colors-- both terms are divisible by 4 and a squared.
- So let's fact out a 4a squared.
- So we get 4a squared.
- 12 divided by 4 is a 3.
- And a cubed divided by a squared is an a.
- So 12a cubed divided by 4a squared is 3a.
- Minus 20-- I could say plus minus 20--
- but you get the idea.
- 20 divided by 4 is 5.
- And a squared divided by a squared is just a.
- And if you don't believe this, multiply it out.
- 4a squared times 3a is 12a cubed.
- And 4a squared times minus 5 is minus 20a squared.
- So it works out.
- You do the denominator.
- Let's see, both of these are divisible by 8a, so let's
- factor that out.
- 16 divided by 8 is 2.
- a squared divided by a is a.
- So 16a squared divided by 8s is 2a.
- And if you go the other way, 8a times 2a
- squared is 16a squared.
- So it all works out.
- Plus 1.
- 8a times 1 is 8a.
- So let's see what we could do here.
- This is becomes a 1.
- This becomes a 2.
- And a squared divided by a, this becomes a 1 and this just
- becomes just an a.
- And we're left with a times 3a minus 5, over 2
- times 2a plus 1.
- And let's see.
- That is choice D.
- I thought maybe they'd want us to re-multiply this out again.
- But that is choice D.
- Problem 68.
- Oh this is a good one.
- I'll just write it.
- They want us to multiply something.
- So they say 7z squared plus 7z-- all of that--
- over 4z plus 8.
- Times z squared minus 4-- all of that-- over z to the third
- plus 2z squared plus z equals.
- So you must be like oh my god, I have to multiply all of
- these things and I have to divide them.
- But the best thing, I'm guessing, is to just factor
- these out and all sorts of things will start canceling
- out with each other.
- And it will turn into a pretty simple problem.
- Let's see, both of these terms are divisible by 7z.
- So le'ts factor that out.
- So that top part becomes, 7z squared divided by 7z, you
- just have a z left.
- If multiply these, you get 7z squared.
- Plus 1.
- If you multiply this out, you get 7z squared plus 7z.
- When you multiply fractions, it's just the numerator times
- the numerator, over the denominator times the
- denominator.
- So this is times the numerator.
- Z squared minus 4, that's a squared minus b squared.
- So that's z plus 2, a plus b, times z minus 2, a minus b.
- That's just the pattern when I say all those a's and b's.
- So that's z plus 2 times z minus 2, hopefully you can
- recognize that at this point.
- And then all of that over-- let's see, we can definitely
- factor out a 4 here, so that's 4 times z plus 2.
- 8 divided by 4 is 2 times-- so we can definitely factor out a
- z here, so we get z times z squared plus 2z plus 1.
- I think we're almost done.
- Now we have to factor this.
- Let me just rewrite everything.
- So this is equal to 7z times z plus 1, times z plus 2,
- times z minus 6.
- All of that over 4 times z plus 2, times z.
- And what's this?
- This is z plus 1 squared.
- Z plus 1 times z plus 1, 1 times 1 is 1,
- and 1 plus 1 is 2.
- So times z plus 1, times z plus 1.
- And now is fun part.
- This is a 1 here, that's parentheses.
- Now we can start canceling out.
- And we assume that the denominator would never equal
- 0 and all that.
- Let's see, this z plus 2 cancels out
- with this z plus 2.
- This z plus 1 cancels out with one of these z plus 1's.
- I'll do the one that's written messier.
- And let's see, this z cancels out with this z.
- And what are we left with?
- Everything simplified to 7 times z minus 6 over 4
- times z plus 1.
- I wrote a z minus b here.
- It's z plus 2 times z minus 2.
- All that pattern matching, I made a mistake.
- Z squared minus 4 is z plus 2 times z minus 2.
- Not z minus b, and I though that was a 6.
- So this is z minus 2.
- So this is z minus 2.
- And so that is choice A.
- Sorry about that error.
- Brain malfunctions all the time.
- All right, now they want us to do it again.
- They want us to find the product of x plus 5, over 3x
- plus 2, times 2x minus 3, over x minus 5.
- Frankly, there's not a lot of simplification we can do, we
- just have to multiply it out.
- So this is going to be equal to x plus 5 times 2x minus 3.
- All of that over 3x--
- I'm just multiplying the numerator and then multiplying
- the denominators-- 3x plus 2 times x minus 5.
- And now we just multiply both binomials, x
- times 2x, 2x squared.
- X times minus 3, minus 3x.
- 5 times 2x, plus 10x.
- 5 times minus 3, minus 15.
- Fair enough.
- Now you do the denominator.
- 3x times x is 3x squared.
- 3x times minus 5, minus 15x.
- 2 times x, plus 2x.
- 2 times minus 5, minus 10.
- And now let's see if we can simplify.
- We have the numerator is equal to 2x squared
- minus 3x plus 10x.
- So that's plus 7x minus 15.
- All of that over 3x squared.
- And minus 15x plus 2x.
- That's minus 13x minus 10.
- And that is choice D.
- Next problem.
- Problem 70.
- Boy, they they want us to keep this up.
- This is good practice.
- So they write, x squared plus 8x plus 16, over x plus 3,
- divided by 2x plus 8, over x squared minus 9.
- So the first thing you do, when you divide by a fraction,
- it's the same thing is multiplying by its inverse.
- So this is equal to x squared plus 8x plus 16, over x plus 3
- times the inverse of this, x squared minus 9,
- over 2x plus 8.
- Fair enough.
- Now let's see if we can simplify these a little bit.
- I'll do that in yellow.
- So this is, 4 plus 4 is 8, 4 times 4 is 16.
- So this we can rewrite as x plus 4 times x plus 4.
- x squared minus 9, that's a squared minus b squared.
- So this we can rewrite as x plus 3 times x minus 3.
- It's going with the pattern.
- We can factor out a 2 here, so we can rewrite this as 2
- times x plus 4.
- We have an x plus 3 there.
- And of course, when we multiply fractions, we're just
- multiplying all the numerators over all the denominators.
- So it's almost like you make this one line.
- So the numerator is x plus 4 times x plus 4 times x plus 3
- times x minus 3.
- All of that over x plus 3 times 2 times x plus 4.
- So now let's do some cancellation.
- This is the fun part.
- So we have an x plus 4 and and x plus 4, cancel them out.
- We have an x plus 3 and an x plus 3, cancel them out.
- And what are we left with?
- We are left with an x plus 4 times an x minus 3.
- All of that over 2.
- And that is choice C.
- And I will see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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