I have a month to learn college algebra

[Risha]

@Indistinguishable

Sure; my wording there was kind of clumsy. What I was saying was that you take the polynomial P with variable x, and you generate from it a new polynomial Q by replacing all the instances of x with x+1 instead. So, with our particular example of P(x) = 2x^3 + 3x^2 + 5x + 6, we would have that Q(x) = 2(x+1)^3 + 3(x+1)^2 + 5(x+1) + 6 [which can be expanded/"FOIL"ed out, but there’s no need to do so].

Clearly, this construction causes Q(0) to equal P(1), Q(3) to equal P(4), etc. In particular, Q(9) will equal P(10), which is the number we originally wrote in decimal.

Did that help?

That it did! The part in bold was the link I was missing. Thanks.

(I’m just glad that I was able to follow everything else without help. It really has been a long time since I’ve used any of this. )

[Indistinguishable]

@Spectre of Pithecanthropus

I worked it out myself once, but I had two supporting lemmas and the whole thing took nearly two pages. Like I said, I’m not that good at math but I find some aspects of it interesting.

Hey, if you proved it for yourself at all, even if it wasn’t the shortest proof, then you’re ahead of the curve, I think.

Next, let’s show the OP how to prove the volume of a sphere.
d&r…

I would be immensely pleased if anyone found a nice way to do so with college algebra alone.

[Cisco]

I did the first of a set of sample tests and got 5 out of 6 correct. I don’t understand the one I got wrong, could someone please explain it to me? It was:

-2|3-4-5|=?

A. -12
B. -8
C. 8
D. 12
E. 24

I came up with B. -8, but the correct answer was A. -12.

I know the || around a number means absolute value and it sort of cancels out the negative sign if there is one, but I don’t know how it’s supposed to be handled with multiple numbers. I think what confused me is that I couldn’t figure out what the relationship between the 2 and the 3 was. Was I supposed to add them, multiply, ?
( . . . pondering . . . )
I think I figured it out, if I’m supposed to treat the || like parentheses, but I’ve never seen them used like that before.

[Thudlow_Boink]

@Cisco

I think I figured it out, if I’m supposed to treat the || like parentheses

Yes, that’s right.

[Indistinguishable]

@Thudlow Boink

Yes, that’s right.

No, that’s not right. (If they were parentheses, the answer would be D. 12, not A. -12).

@Cisco

I know the || around a number means absolute value and it sort of cancels out the negative sign if there is one, but I don’t know how it’s supposed to be handled with multiple numbers. I think what confused me is that I couldn’t figure out what the relationship between the 2 and the 3 was. Was I supposed to add them, multiply, ?

Yes, they act as absolute value here too. The multiple numbers within it are an expression “3-4-5”. Calculate the value of that (three minus four, minus five, which equals -6), and then take its absolute value (which is 6).

The relationship between the -2 and the bit within the absolute value bars is that they should be multiplied (thus, the final answer is -2 times 6, which is -12). In general, when two quantities are next to each other (and not just the separate digits of a multi-digit number or something like that), this means they are to be multiplied (e.g., “3y” means “3 times y”).

[Indistinguishable]

@Cisco

I came up with B. -8, but the correct answer was A. -12.

It looks like your mistake was parsing “3-4-5” as 3 - (4 - 5) [which equals 4] rather than as (3 - 4) - 5 [which equals -6]. The convention is that unparenthesized subtraction “associates to the left” (and similarly for division); i.e., a string of subtractions should be done from left to right, so to speak.

[OpalCat]

It’s like parenthesis except you strip off the negative symbol if the number is negative, right?

[Thudlow_Boink]

@OpalCat

It’s like parenthesis except you strip off the negative symbol if the number is negative, right?

Yes, except that it can be misleading to talk about “stripping off the negative symbol,” because you don’t take absolute values of symbols but of numbers. But the absolute value bars do act as grouping symbols, like parentheses do.

So, for example, | –5 + 2 | is equal to | –3 | (the absolute value of negative 3), which is 3,
not | –5 + 2 | = 5 + 2 (incorrectly “stripping off the negative symbol”).

Also, thinking of absolute value as “stripping off the negative” works if you’re just working with real numbers. If you’re working with other sorts of things, such as complex numbers, or vectors, the absolute value bars have a meaning which is an extension of the real number meaning, and which denotes the “magnitude” of a number, or how far that number is away from zero (without worrying about which direction it is from zero).

[Indistinguishable]

Ah, I see. Thudlow Boink was taking a more sophisticated interpretation of the phrase “treat the | | like parentheses” than I was.

[Thudlow_Boink]

@Indistinguishable

Ah, I see. Thudlow Boink was taking a more sophisticated interpretation of the phrase “treat the | | like parentheses” than I was.

I just figured that, if Cisco had figured out why -12 was the right answer, he must have gotten the right idea of how to treat the | |.

[Cisco]

Here’s one that has me stumped:

If 5 x 10[sup]n[/sup] = 0.005, then n=?

I know the answer is supposed to be -5. I haven’t looked at the key, but I’m sure that’s correct by power of deduction. What am I doing wrong in reaching it, though? Because I keep getting 0.0005, when I know I should be getting 0.005.

10/10 = 1 / 10 = .1 / 10 = .01 / 10 = .001 / 10 = .0001

That’s 10 divided by 10, 5 times, and I get .0001, when I should be getting .001, according to this problem.

[ultrafilter]

You should use logarithms. 5 x 10^n = 0.005 implies that 10^n = 0.001, which implies that log(10^n) = log(0.001). log(a^b) = blog(a), so nlog(10) = log(0.001), and n = log(0.001)/log(10) = -3. For the answer to be -5, you’d need a couple extra zeroes on the right-hand side of the original equation.

[Indistinguishable]

The answer should be -5? Looks to me like the answer should be -3.

At any rate, dividing by 10 five times is the same as multiplying by 10[sup]-5[/sup]; however, you’ve started from 10, so what you produce is 0.0001 = 10 * 10[sup]-5[/sup] = 10[sup]-4[/sup]. The number 10[sup]-n[/sup] is the result of dividing by 10 n many times, but starting from 1. Thus, 10[sup]-3[/sup] is ((1/10)/10)/10 = 0.001, while 10[sup]-4[/sup] = (((1/10)/10)/10)/10 = 0.0001, and 10[sup]-5[/sup] = ((((1/10)/10)/10)/10)/10 = 0.00001.

On edit: Also, yeah, as ultrafilter says, these sorts of problems (solving a[sup]n[/sup] = b for n) are the raison d’etre of logarithms, though, in this case, the numbers are so nice and round that perhaps you needn’t bother.

[Cisco]

Crap. I don’t even know what I don’t know.

[Thudlow_Boink]

@Cisco

Here’s one that has me stumped:

If 5 x 10[sup]n[/sup] = 0.005, then n=?

Actually, this is probably a test of whether you know scientific notation. You can think of the n as how many places you have to move the decimal point.

I know the answer is supposed to be -5. I haven’t looked at the key, but I’m sure that’s correct by power of deduction. What am I doing wrong in reaching it, though? Because I keep getting 0.0005, when I know I should be getting 0.005.

10/10 = 1 / 10 = .1 / 10 = .01 / 10 = .001 / 10 = .0001

That’s 10 divided by 10, 5 times, and I get .0001, when I should be getting .001, according to this problem.

Yes, but you used 10 as your starting point. 5 x 10 would be 50. So, 5 x 10[sup]1[/sup] = 50.
5 x 10[sup]0[/sup] = 5.
5 x 10[sup]-1[/sup] = 0.5.
5 x 10[sup]-2[/sup] = 0.05.
5 x 10[sup]-3[/sup] = 0.005.

Or, if you start with 5 and divide it by 10 3 times, you get to first 0.5, then 0.05, then 0.005. Two more times (five in all) brings you to 0.00005.

[Cisco]

@Thudlow Boink

Yes, but you used 10 as your starting point. 5 x 10 would be 50. So, 5 x 10[sup]1[/sup] = 50.
5 x 10[sup]0[/sup] = 5.
5 x 10[sup]-1[/sup] = 0.5.
5 x 10[sup]-2[/sup] = 0.05.
5 x 10[sup]-3[/sup] = 0.005.

I thought exponents came before multiplication in the order of operations?

And . . . I just wrote that problem down and handed it to my wife. She glanced at it, said “negative 3,” and handed it back. ARRGGGGG!

[Thudlow_Boink]

@Cisco

I thought exponents came before multiplication in the order of operations?

Right. 10[sup]1[/sup] = 10, 10[sup]0[/sup] = 1, 10[sup]-1[/sup] = 0.1, 10[sup]-2[/sup] = 0.01, etc. Each time the exponent goes down by 1, you divide by another factor of 10. (Or, looking at it the other way around, each time the exponent goes up by 1, you multiply by another factor of 10). Then you can multiply 5 x 10[sup]whatever[/sup].

[Indistinguishable]

@Cisco

I thought exponents came before multiplication in the order of operations?

Yes, they do. What appears to indicate otherwise?

[OpalCat]

@Thudlow Boink

Yes, except that it can be misleading to talk about “stripping off the negative symbol,” because you don’t take absolute values of symbols but of numbers. But the absolute value bars do act as grouping symbols, like parentheses do.

Ok, yeah. Well what I meant was “after you’ve done everything inside the parentheses, if the number you get is negative, just take off the -.”

[Risha]

@Indistinguishable

Yes, they do. What appears to indicate otherwise?

I think that this:

Or, if you start with 5 and divide it by 10 3 times, you get to first 0.5, then 0.05, then 0.005. Two more times (five in all) brings you to 0.00005.

was what threw him. We knew that it works and is easy to make sense of because the numbers are so simple, but it does sound like you’re doing it out of order if you’re working straight from the textbook.