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
Algebra II: Imaginary and Complex Numbers 23-26, mostly imaginary and complex numbers
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- We're on problem 21.
- Which product is equivalent to 4x squared minus 16
- over 2 minus x?
- So let's see if we can simplify this a little bit.
- 4x squared minus 16, if we were to factor out a 4, that's
- the same thing as 4 times x squared minus 4.
- That's 4x squared minus 16, all of that over 2 minus x.
- Then x squared minus 4, that's that same pattern that keeps
- showing up, a squared minus b squared, right?
- So that numerator becomes 4 times x plus 2 times x minus
- 2, right? a plus b times a minus b.
- Then the denominator, let's think about it.
- This has 2 minus x, and we have an x minus 2 here.
- But what if we were to factor out a negative 1 from this?
- What happens?
- If you factor out a negative 1, then the negative x becomes
- a positive x, and the positive 2 becomes a minus 2.
- I did that just to show you that a negative, if you divide
- this by or multiply by a negative 1, you
- just switch the order.
- 2 minus x becomes x minus 2, which is a good thing to
- recognize because we have an x minus 2 in the numerator, too.
- So this will cancel out with that.
- If we assume that x does not ever equal 2 because x
- equaling 2 would make this an undefined statement because
- you can't divide by zero, but if we make that assumption, we
- can cancel those out.
- Then we have 4 divided by negative 1 is minus 4
- times x plus 2.
- That is choice D.
- Next question, 22.
- I like this cutting and pasting
- of the actual question.
- It makes it fun.
- I don't want it to slow down my computer too much.
- I sound like I'm melting when that happens.
- So that's our problem.
- I didn't write the choices down.
- All right, let's see what we can do.
- So they have x squared plus 4x.
- Let me write this.
- x squared plus 4x over x plus 3 times x squared minus 9, all
- of that over x squared plus x minus 12.
- You might say, oh, my God, to multiply these using algebraic
- multiplication would take forever, then I'd have to
- factor it and simplify it.
- The trick here is to factor it and simplify it before you do
- any multiplication.
- So let's simplify each of these expressions and cancel
- out as much as possible.
- So here we can just factor out an x.
- So this just becomes x times x plus 4.
- This is a squared minus b squared, so this becomes x
- plus 3 times x minus 3.
- With that you can't do much.
- Let's think about this one.
- What two numbers add up to a 1, a 1x, and when you multiply
- them, you get a minus 12?
- So let's see, if you have an x plus 4 times x minus 3, that
- should work.
- 4 minus 3 is 1.
- 4 times minus 3 is minus 12.
- So that works.
- All right now we're ready to cancel.
- Normally.
- I would just start canceling, but I just
- want to make the point.
- So this is equal to x times x plus 4 times these, x plus 3
- times x minus 3, all of that over x plus 3 times x plus 4
- times x minus 3.
- Now we can cancel.
- We have an x minus 3, x minus 3, x plus 4 x plus 4, x plus
- 3, x plus 3, and all we're left with is an x.
- So all of this fancy expression, all that
- equals is an x.
- That's choice B.
- Where am I?
- Next problem.
- Image, clear image, invert colors, there you go.
- All right, let me see if I have to copy
- and paste this one.
- Where was I?
- Oh, why not?
- Another one where they're trying to get
- the simplest form.
- So I copied it.
- Let me paste it.
- All right, I hope you can read this.
- So what is the simplest form of this expression here?
- I'll rewrite it.
- 5x to the third y plus 20x squared y squared plus 20x y
- to the third, everything divided by 5xy.
- So 5xy actually goes into every one of
- these terms, right?
- Let's just get rid of the 5's.
- Well, actually, let's just divide the numerator and the
- denominator by 5xy.
- So what's 5x to the third y divided by 5xy?
- Well, 5 divided by 5-- I'll do it in magenta.
- 5 divided by 5 is 1 so we don't have to write that.
- x to the third divided by x is x squared.
- y divided by y is 1.
- So this term divided by this term reduced to that.
- Easy enough.
- Next one.
- Plus 20 divided by 5 is 4.
- x squared divided by x is x.
- y squared divided by y is y.
- We did the second term.
- Let's do the third term.
- 20 divided by 5 is 4.
- x divided by x is 1 so we don't have to write it. y to
- the third divided by y is y squared.
- What we have here is this simplified, but that doesn't
- look like anything here.
- So my guess is that this is probably one of these.
- This is the square of something.
- Let's see if we can factor this.
- This would be x.
- So what two numbers when I add them become 4xy or what number
- when I multiply it times x is equal to half of this-- that's
- another way to think about it-- and when I square it is
- equal to 4y squared?
- That's just plus 2y squared.
- If that didn't make a lot of sense, multiply this out and
- you'll get this answer.
- When you look at the choices, you should just say, OK, I
- know that this choice and this choice, those look
- nothing like that.
- Then you say, oh, well, maybe this is one of
- these choices, right?
- But you have a 4y squared here.
- This one has no y at all in it.
- So you can even just use deductive reasoning to say,
- oh, the choice must be x plus 2y squared.
- Next problem.
- OK, this one I'm definitely going to copy and paste
- because they have a big image.
- All right, let me see if I can do it.
- This makes it a lot easier than having to draw it myself.
- All right.
- If i equals the square root of negative 1, which, if you
- didn't know it, that's what i is defined as, and I go on
- many videos about the magical of i and e and pi.
- Which point shows the location of 5 minus 2i on the plane?
- The x-axis is the real axis.
- You take the real part, which is 5.
- So you go 1, 2, 3, 4, 5.
- So that's the real part.
- Then the y-axis is the imaginary axis.
- You take the imaginary part, minus 2i, minus 2, right?
- So you go down 2 in the i direction and you're at point
- D, which is choice D.
- Fair enough.
- That problem had a lot of graphics but
- was unusually simple.
- Next problem.
- OK, now they're giving us a whole parade of imaginary
- number questions.
- So this one I don't have to cut, copy, and paste.
- They say if i is equal to the square root of negative 1,
- which is a definition that we've all learned, what is the
- value of i to the fourth?
- Well let's just go back.
- Let's say you've never seen i before in your life.
- Let's think about it.
- If i is equal to negative 1-- sorry, this is negative 1.
- I know that negative looks like it got messed up.
- Let's square both sides of this equation.
- i squared will be equal to negative 1.
- Now, let's square both sides of this equation.
- i squared squared is equal to negative 1 squared.
- Well, i squared squared is just i to the fourth, right?
- 2 times 2 is equal to negative 1 squared, which is
- just equal to 1.
- And we're done.
- That's choice C.
- All right, now they have another imaginary graph.
- So let me just copy and paste this graph.
- I'll go here.
- There you go.
- I didn't have to erase what I had before.
- The question they ask us: Which of the following complex
- numbers is represented by the point on the graph below?
- So we just have to figure out the real and
- the imaginary parts.
- So what's the real part?
- This coordinate, its real part is 3.
- Its imaginary part is minus 4 right there.
- So it's 3, and its imaginary part, minus 4 in
- the imaginary direction.
- And we're done.
- 3 minus 4i.
- That's choice C.
- I'm almost out of time.
- 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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