Algebra II
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California Standards Test: Algebra II
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California Standards Test: Algebra II (Graphing Inequalities
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CA Standards: Algebra II (Algebraic Division/Multiplication)
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CA Standards: Algebra II
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Algebra II: Simplifying Polynomials
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Algebra II: Imaginary and Complex Numbers
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Algebra II: Complex numbers and conjugates
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Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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Algebra ||: Conic Sections
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Algebra II: Circles and Logarithms
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Algebra II: Logarithms Exponential Growth
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Algebra II: Logarithms and more
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Algebra II: Functions, Combinatorics
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Algebra II: binomial Expansion and Combinatorics
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Algebra II: Binomial Expansions, Geometric Series Sum
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Algebra II: Functions and Probability
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Algebra II: Probability and Statistics
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Algebra II: Mean and Standard Deviation
CA Standards: Algebra II (Algebraic Division/Multiplication) 9-12 (algebraic division and multiplication)
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- All right.
- We're on problem number 9.
- At least for the next few, I don't think I have to copy and
- paste these problems. I can just write it.
- It looks like a little bit of algebraic long division.
- Problem 9: They want us to divide 2x plus 7 into-- it's
- pretty long so I'll give it some space-- 2x to the fourth
- plus 21x to the third plus 35x squared minus 37x plus 46.
- This might look daunting, but once you
- start, it's not so bad.
- So like any long division, you kind of look
- at the biggest place.
- And the largest degreed place, at least in this polynomial,
- is the 2x place.
- And that's what we work with.
- So we say, how many times does 2x go into 2x to the fourth?
- Well, let's see.
- The 2 goes into the 2 one time, and x goes into x to the
- fourth x to the third times.
- What's x to the fourth divided by x?
- It's x to the third.
- So we say it goes into it x to the third times.
- And if we were right, when we multiply, we should get 2x to
- the fourth.
- So x to the third times 2x is 2x to the fourth.
- And then x to the third times 7 is plus 7x to the third.
- If this was some other-- well, we'll worry about it when we
- get to it, but you would want to put it under the
- appropriate place in this number.
- It just happened to be that x to the third was the next one.
- Now you subtract just like you do with regular long division.
- These cancel out.
- 21 minus 7 is 14x to the third.
- And then you could bring down all of this, but we can worry
- about that in a second.
- So how many times does 2x go into 14x to the third?
- Well, 2 goes into 14 seven times, plus 7, and x goes into
- x to the third x squared times.
- If we're right, when we multiply, it should work out.
- So 7x squared times 2x is equal to 14x to the third,
- like we thought it should.
- 7x squared times 7 is what?
- That's plus 49x squared.
- And now we subtract.
- 14x to the third minus 14x to the third.
- That's 0.
- Now we have 35-- we could bring this down, just because
- it makes it easier.
- 35x squared.
- 35x squared minus 49x squared.
- Let's see.
- So it's minus 14, right?
- 49 minus 35.
- So it's minus 14x squared.
- And then 2x goes into minus 14x squared.
- So 2 goes into minus 14 minus 7 times, and then x goes into
- x squared x times.
- When you multiply this minus 7x times 2x, so you get minus
- 7 times 2 is minus 14.
- x times x is x squared.
- Of course, you have to multiply it
- times both terms, right?
- We're multiplying it times this whole expression.
- Minus 7x times 7 is minus 49x.
- Now we're ready to subtract again and I'm
- running out of space.
- These cancel out.
- We could bring this minus 37x down, so minus 37x.
- Remember, we're subtracting this, right?
- We could almost add the negative.
- Let's make the negative.
- So this is minus 37x minus minus 49x is the same thing as
- minus 37x plus 49x.
- I made both of those positive just to
- simplify it in my brain.
- Oh, whoops.
- I didn't want to go down that much.
- OK.
- So these cancel out and then this is equal to 12x.
- And 2x plus 7 goes into 12x.
- Well, now, let's see.
- 2 goes into 12 six times, x goes into x one time, so then
- this is equal to plus 6.
- 6 times 2x plus 7.
- 6 times to 2x-- I know you can't see it-- is 12x.
- And then 6 times the 7 is 42.
- We have a 46 up there.
- I should have written smaller.
- So you get this 46.
- And now we're ready to subtract again.
- 12x minus 12x is 0.
- 46 minus 42 is 4, so we have a remainder of 4.
- So let's see, our answer is all of this
- with a remainder 4.
- So all of that plus the remainder over what we're
- using to divide into: 2x plus 7.
- So our answer, let's see what choice it is.
- It's x to the third plus 7 x squared minus 7x plus 6 plus 4
- over 2x plus 7.
- We just took our remainder and said, OK, whatever this is,
- that's also there.
- So that is choice-- let's see, x to the third plus 7x squared
- minus 7x plus 6 plus 4 over 2x plus 7.
- So it's actually choice d is our answer.
- I was just reading choice a.
- I should have shown you that.
- So choice d is our answer.
- Next question.
- Problem number 10.
- That one took up more space than I expected.
- Problem number 10.
- Which polynomial represents-- so they want us to multiply--
- this how they write it-- 3x squared plus x minus 4 times
- 2x minus 5.
- I find that we should just multiply these the way I would
- multiply any number.
- So I would rewrite it as 3x squared plus x minus 4 times,
- and I put the same degree under the same degree, times
- 2x minus 5.
- And now let's take the 0th degree place.
- I guess you could view that way.
- Minus 5 times minus 4 is plus 20.
- Minus 5 times x minus 5x.
- Minus 5 times 3x squared is minus 15x squared.
- 2x times minus 4 is minus 8x.
- I'll put it in this place because that's our x's space.
- 2x times x is 2x squared.
- And then I should have had all of this-- actually, I think I
- can do that.
- Let me see if I can move all of this to the right.
- The magic of computers.
- I can move it all to the right, although I should color
- that back in.
- Let's see if I can color that.
- Look at that!
- All right, back in business.
- All right.
- So I did 2x times minus 4 is minus 8x.
- 2x times x is 2x squared.
- 2x times 3x squared is 6x to the third.
- This is a plus.
- Now we just add up everything.
- So we get 6x to the third minus 15 plus 2 minus 13x
- squared minus 13x plus 20.
- So 6x to the third minus 13x squared-- I'm just checking
- all the choices-- minus 13x plus 20.
- That's choice b.
- Problem 11.
- Problem 11.
- Let's see, they wrote-- this is problem 11, just so you
- know what we're doing.
- Minus 2x squared plus 6x plus 1 minus 2 times 4x squared
- minus 3x plus 1 is equal to-- well now, we just have to
- simplify this.
- Let me rewrite this part.
- This is minus 2x squared plus 6x 1.
- Let's distribute a minus 2.
- So minus 2 times 4x squared is minus 8x squared.
- That's that.
- Minus 2 times minus 3x, that's plus 6x.
- Minus 2 times 1 is minus 2.
- And now, let's see.
- We can add the like terms up.
- Let's see what this is equal to.
- Let's do the x squared term.
- So first I'll do the minus 2x squared and minus 8x squared,
- so that is minus 2 minus 8, that's minus 10x squared.
- Let me do another color for each of them.
- Then I have the x terms. I have 6x, and then I have
- another 6x.
- You add them together, you get 12x, plus 12x.
- And then you have your ones terms or x to the 0th power,
- you can view it.
- You have plus 1 and minus 2, so that's minus 1.
- And then that is choice-- let's see, let me scroll down.
- That is choice d. minus 10x squared plus 12x minus 1.
- Choice d.
- Problem number 12.
- Problem 12.
- We're on a roll.
- Problem 12: Which expression is equivalent to-- so now
- they're multiplying.
- They write 6y squared minus 2 times 6y plus 2.
- You could do this in your head, and you should practice
- doing it in your head, but if you never want to make a
- mistake, you just write it like this.
- 6y squared.
- And I have nothing in the x's spot.
- And I say minus 2.
- And then this one is 6y.
- I write that in the y spot.
- This is y squared, this is y, this is y to the 0, which is
- just 1, plus 2.
- And now I multiply them.
- 2 times minus 2 is minus 4.
- 2 times 6y squared is 12 y squared.
- So I write that in that spot, 12y squared.
- Now 6y times minus 2 is minus 12y.
- So that'll go here.
- And then 6y times 6y squared is 36y to the third.
- So we have our answer.
- It is-- you just add them up-- 36y to the third plus 12y
- squared minus 12y minus 4.
- And that is choice d.
- And you could do this in your head, and that's a good thing
- to practice.
- But I just wanted to make sure it was clean when I did it.
- And let's see, should I continue with the next video
- or should I stop there?
- Because I'm at 11 minutes.
- I'll stop there and I'll see you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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