TheXSquaredFactor

It is not a question, though.

I hope this helps!

helperid1839321, I decided to delve deeper into this subject. You may like my findings!

First of all, I think I understand why this property occurs. I happen to know that \(142857\) is the first 6 digits of the decimal expansion of \(\frac{1}{7}\). At first, I thought this was insignificant, but it turns out that this fact can be used to understand this further. Look at the table below.

This forced me to wonder if there is any more solutions for \(\frac{1}{p}\), where p is a whole number that creates this special property. This is because if another number p causes some repetition, then we would have found another number! YAY!

I decided to enlist some help from a computer here. This is what the computer outputted. 

7, 17, 19, 23, 29, 47

WHAT! There are more! Yes, these have the same property. Let's check them out, shall we?

It appears as if 142857 is the only number that does not start with a zero. Let's keep running this simulation! Thankfully, more numbers output! I let it run for some time, too. 

59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193...

In case you are wondering,

 \(\frac{1}{193}\)=005181347150259067357512953367875647668393782383419689119170984455958549222797927461139896373056994818652849740932642487046632124352331606217616580310880829015544041450777202072538860103626943...

There are a few patterns that I see here 

1) \(p\) must be a prime number

2) The decimal expansion must have a maximum period decimal expansion of \(p-1\).

I am also making a conjecture here that I do not know whether or not is true: there are an infinite number for p that create these types of numbers!

The triangle sum theorem states that the sum of the measures of the interior angles of a triangle equals 180 degrees. Using this theorem alone, one can find the measure of all the angles.

Now, plug in this value for the angle measure expressions.

\(m\angle A=x-16=98-16=82^{\circ}\)

\(m\angle B=2x-155=2*98-155=196-155=41^{\circ}\)

\(m\angle C=\frac{1}{2}x+8=\frac{1}{2}*98+8=49+8=57^{\circ}\)

1990 nearly lies in the middle of the years 1972 and 1997. Since it is given that the amount of milk produced can be modeled using a linear model, I would expect the approximate amount of billions of pounds of milk would be the average of the amount produced for those given years.

Let m = milk produced (in billions of pounds)

\(m\approx\frac{7+8}{2}=\frac{15}{2}=7.5\)

Therefore, the third answer choice is correct.

I would have used synthetic division at the final step when I checked whether or not x+1 is a factor of the remaining polynomials, but trying to line everything up was a too arduous task for me...

I understand your confusion. Saying that there is one solution is correct. Saying that x=4 and x=10 are both solutions is incorrect. I would put "There is only one solution: x = 10." I believe this to be the most correct.

For some strange reason, though, it is listed twice in the given options.

In order to figure this one out, we will utilize some bits of manipulation.

The y-intercept of a polynomial function is determined by the constant term of each polynomial. We know, however, that the y-intercept is located at \((0,-5)\). Knowing this, we have already eliminated the last answer choice, \(y=x^4+4x^3-2x^2-5x\) because its y-intercept is located at the origin. 

The next rule I will use is something called the Descartes' rule of signs. We know, based on the given info, that at least one zero of this polynomial is negative. 

For the three remaining candidates, let's plug in the function \(f(-x)\) and count the number of sign changes. If you don't mind, I converted all the equations to function notation. 

Why am I doing this, you may ask? Well, the Descartes' rule of signs states that the number of sign changes in \(f(-x)\) equals the amount of negative zeros a function has or less than by an even number. 

We can make a table to show to show the amount of negative zeros that each function can have. I'll put it in table format for your convenience.

Wow, we have eliminated \(y=x^3-4x^2+2x-5\) from the candidates because it has no real negative roots. However, this is even more helpful than it may appear at first. Now, we will finally plug in our first point, \((-1,0)\) and see if results in a true statement. I'll choose \(x^3+4x^2+2x-5\). 

This is very useful information. I now know that \((-1,0)\) does not satisfy \(y=x^3+4x^2+2x-5\), which means that this point cannot lie on this line. By process of elimination, this means that \(y=x^3+4x^2-2x-5\), the second answer choice from the top, is the correct equation.

To reference segments as figures, it is considered standard to use an overbar to make this clear.

\(\overline{AB}\) means a segment

\(AB\) means the length of the segment. 

Get it?

Ok, thanks for replying. One way to check if your solution is right is to plug it in and see if the resulting equation is indeed a true one. Let's do that!

\(\sqrt{2x-4}-x+6=0\)

Let's plug in x=4 and x=10 and see if we get a true statement.

Check x=4

Check x=10

Ok, I have verified that x=10 is a valid solution, but x=4 does not satisfy this equation. I can tell you that you are wrong. The only issue here is that I cannot determine if x=4 is indeed an extraneous solution or not. I'll just solve the equation.

Ok, you solved the equation correctly, but x=4 is an extraneous solution. However, there is no option choice that states both that x=4 is a solution and x=10 is an extraneous one.