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 13-16 (polynomial factoring and multiplication)
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- I copied and pasted the next view because they had a little
- bit of a drawing, so I thought, better than me
- drawing it myself.
- So we're on problem number 13 and I cut it off a little bit.
- What is the volume of the figure below?
- Well, the volume is just the width times the
- height times the depth.
- So the volume is just going to be-- the width is x plus 4
- times x plus 1 times x plus 6.
- X plus 4 times x plus 1.
- We can do this in our head, but I always find it easier to
- do it this way. x plus 1 times x plus 1.
- 1 times 4 is 4.
- 1 times x is x.
- And then you have the x place. x times 4 is 4x, x
- times x is x squared.
- Fair enough.
- So that equals x squared plus 5x plus 4.
- Now, we have to multiply that times x plus 6, right?
- We did x plus 4 and x plus 1, and this was a 6. x plus 6,
- that's this side.
- So now we multiply this times x plus 6.
- 6 times 4 is 24.
- 6 times 5x is 30x.
- 6 times x squared is 6x squared.
- I hope you can read what I'm writing.
- Scroll down a little bit.
- Now we do the x's place.
- I'll do it in a different color because I think I'm
- going to run out of-- so x times 4 is 4x.
- x times 5x is 5x squared.
- x times x squared is x to the third.
- Have to run into the white area.
- So this is x to the third.
- What's 6 plus 5?
- Plus 11x squared plus 30x-- no, no, sorry.
- 30 plus 4 is plus 34x plus 24.
- So x to the third, all of these choices are x to the
- third, plus 11x squared, so it's either b
- your d, plus 34x.
- That's that one.
- Plus 34x plus 24.
- So the choice is b.
- Next problem All right.
- 8a to the third plus c to the third.
- Which of these choices?
- So that's interesting.
- This is not a trivial thing to just factor.
- We'll have to do some intuition here.
- Well, let's think about it.
- In order to have a positive number for the a term, every
- time we're multiplying, we have to have a positive, so
- this is 2a times 2a times 2a.
- That would get us to 8a to the third.
- But there's something interesting about choice a,
- why I already feel like canceling out choice a.
- Choice a is essentially 2a plus c to
- the third power right?
- Choice a is 2a plus c to the third power.
- And every time you multiply this out, every time you
- multiply 2a plus c times itself, you're going to add
- more terms. No terms are going to cancel out.
- I mean, if you were to square this, you're going to get,
- just to show you, you get 4a squared
- plus 4ac plus c squared.
- And then to get the third power, you'd multiply that
- times 2a plus c.
- So you're going to get this thing.
- You're actually going to get four terms. Choice a is
- definitely not the case.
- In order for one of these to simplify to this, we're going
- to have to have some positives and negatives there to cancel
- out terms as you expand these polynomials.
- So let's look at the other choices and see if
- they make any sense.
- 2a times-- and I like looking at the a terms and what
- generates the c cubed terms and maybe we can cancel out
- some things.
- 2a times 4a squared, that's 8a to the third.
- But minus c times c squared, that's minus c to the the
- third, right?
- So I don't even have to worry about multiplying
- this whole thing out.
- I know because the c cubed term is going to come only
- from this term multiplied by this term.
- There'll be other terms that hopefully will cancel out, but
- we know that choice b cannot be the choice because minus c
- times c squared is minus c to the third, and this is a plus
- right here.
- So we know b can't be the right answer.
- Let's see, choice c.
- I can use the same logic.
- We have a minus c and we have a plus c.
- We have a plus c here.
- We would have to have two plus c's in order for this to work,
- so this can't be the answer.
- So just by deductive reasoning, I can say it's
- probably choice d, but I'm going to prove it to you since
- we're here to learn, not to just answer things correctly.
- So if I were to multiply those two numbers, I have 4a squared
- minus 2ac plus c squared, and I'm multiplying that
- times 2a plus c.
- So I'm going to mix up all of the terms a little bit, but
- you will get the idea.
- So the important thing when you're multiplying, you could
- do this in your head, is that you multiply each of these
- terms times each of those.
- So c times all of that plus this times all of of that.
- So c times all of that is c times c squared.
- I'm not doing it in the right place notation like I did last
- time because it gets a little more confusing now.
- But c times c squared is c to the third.
- c times minus 2ac is minus 2ac squared.
- c times 4a squared is 4a squared c.
- And now we can do this one.
- 2a times c squared.
- I'm going to put it in this place because it turns into ac
- squared, right? ac squared and I have a 2.
- So 2a times c squared is 2ac squared.
- 2a times minus 2ac, you get a minus 4a squared c, so it's
- minus 4a squared c.
- This is a plus here.
- And then 2a times 4a squared is 8a to the third.
- And luckily enough, let's see, they get 8a to the third.
- This cancels out, this cancel out, and you get
- plus c to the third.
- So we were right by doing our deductive reasoning, that
- choice d was the answer.
- Next problem.
- I've run out of space.
- Let me see what the next problem even is.
- Actually, where was I?
- OK, I see.
- OK, problem 15.
- The total area of a rectangle is 4x to the
- fourth minus 9y squared.
- Which factors could represent the length times the width?
- OK, so essentially you're saying the length times the
- width is going to be equal to the area, which is 4x to the
- fourth minus 9y squared.
- So to some degree, we have to factor this into multiple
- expressions.
- And the first thing you should always look at when you have
- to factor any type of an expression, especially if
- there's only two terms in the expression, does it fit the
- form a squared minus b squared?
- Because if it does, you can factor this into a plus b
- times a minus b.
- And if you don't trust that these equal each other, I've
- made multiple videos about it, but just multiply them. a plus
- b times a minus b is equal to a squared minus b squared.
- And luckily, this fits that format, right?
- If a squared was equal to 4x to the fourth, then what's a?
- a would be equal to 2x squared.
- And if b squared is equal to 9y squared, then b is equal to
- the square root of that, which is 3y.
- And so this definitely is equal to a
- squared minus b squared.
- So this expression right here can be factored into a plus b
- times a minus b.
- So a plus b is 2x squared plus 3y times a minus b.
- 2x squared minus 3y.
- Let's see, which choice is that?
- 2x squared plus 3y?
- OK, they switched the order.
- Choice a says 2x squared minus 3y times 2x squared plus 3y,
- which is the same thing.
- You can switch orders when you're multiplying, so the
- answer is a.
- Next problem, problem 16.
- I'll switch colors arbitrarily.
- Problem 16: Which product of factors is equivalent to-- let
- me copy and paste this because this is an interesting-- so
- let me copy and paste this problem.
- OK, I've copied it, and now let me paste it here.
- OK, which product of factors is equivalent to that thing
- over there?
- So let's simplify that.
- That's my first impulse.
- No, no, no, we don't even have to simplify it.
- This fits our same pattern, right, if we can fit the
- pattern a squared minus b squared.
- So look at this thing.
- This has x plus 1 squared minus y squared.
- So if we said that a is equal to x plus 1 and b is equal to
- y, then we fit the pattern.
- This is a squared minus b squared.
- So then this thing can just factor into a plus b
- times a minus b.
- So what's a plus b?
- It's x plus 1 plus y.
- That's a plus b.
- And then a minus b is x plus 1 minus y.
- And that is choice d.
- That problem was easier than I thought I was going to be.
- Choice d, right there.
- And I'm out of time again.
- 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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