TheXSquaredFactor

In the rational function \(f(x)=\frac{2x}{x^2-5x-14}\), we can determine the vertical asymptotes by setting the denominator equal to 0 and solving for x. 

The horizontal asymptote requires some observation. The horizontal asymptote lies on y=0 since the degree of the numerator is less than the degree of the denominator. Therefore, c=0. We know the unknown variables, so we can now calculate their collective sum.

\(a+b+c\\ 7-2+0\\ 5\)

Let's call the number that one is thinking of "x."

If I take "x" and I multiply "x" by 3, then I have "3x."

If I take "3x" and I add "4," then I have "3x+4."

If I have "3x+4" and I divide by 4, then I have \(\frac{3x+4}{4}\).

If the original number is "x," then three less than "x" is x-3.

We know that \(\frac{3x+4}{4}\) and \(x-3\) are equivalent because of the keyword "is." 

Therefore, the final algabraic equation would be \(\frac{3x+4}{4}=x-3\)

a) In order to factor using this method, let's try and identify the GCF first. 9 is the greatest common factor between 36 and 81. a^4 is the greatest common factor between the a's, and b^10 is the factor for the b's. Let's factor it out!
 

b) The beginning binomial is a difference of squares to begin with, so it is possible to start with this first! 

Well, these are the two techniques. 

8) This one will require the formula that yields the volume of a cylinder. \(V_{\text{cylinder}}=\pi r^2h\). We can manipulate this formula so that we can find any missing information such as the height, in this case. 

9) If the height of the un-consumed soup was 8 centimeters tall and 3-centimeters-worth of soup is consumed, then 5 centimeters of soup remains. We already know the radius of this soup can (that I assume is cylinder-shaped despite not being explicitly stated), so we can determine the volume.

10) If the town park enlarges its area by a factor of 5, then both dimensions of the park are affected by this scale factor. For example, if we assume, for the sake of understanding, that the park is perfectly rectangular with dimensions 3yd by 100yd, then both dimensions (the length and the width) would be affected by this scale factor. This means that, on area, the scale factor actually affects the area by its square, or 25 in this case. 

\(300\text{yd}^2*25=7500\text{yd}^2\)

The problem is easier than I first thought. 

By the given information, we know that there are three right-angled triangles in the diagram. We know that \(m\angle AEB=m\angle BEC=m\angle CED= 60^{\circ}\). We can use this information to determine that every right triangle is also a 30-60-90 triangle. We also know that \(AE=24\).

A 30-60-90 triangle is a special kind of right triangle where the ratio of the side lengths are \(1:\sqrt{3}:2\). \(\overline{AE}\) is the longest side length because it is the hypotenuse of the largest right triangle in the diagram. \(\overline{BE}\) is the shortest side length of \(\triangle ABE\) because this side is opposite the smallest angle. We mentioned earlier what the ratio of the side lengths are, so we can determine the length of \(\overline{BE}\) without doing anything too computationally demanding. 

Of course, the ultimate goal is to figure out the length of \(\overline{CE}\). If you look at \(\triangle BCE\), carefully, you will notice that we are in an identical situation to when we solved for \(BE\). Notice that \(\overline{BE}\) is the hypotenuse of this triangle, and \(\overline{CE}\) is the shortest side length since it is opposite the 30º angle. We can use the same \(1:\sqrt{3}:2\) relationship of the side lengths to find the missing length.

This question requires one to understand certain features of a rational function in order to determine the horizontal asymptote. 

#1) The degree of the numerator is 6, and the degree of the denominator is 5. Since the degree of the numerator exceeds the degree of the denominator, no horizontal asymptote exists for the first function.

#2) There is a general process to graphing rational functions. 

1) Factor the numerator and denominator completely, if possible

In this case, no factoring can be done to either the numerator or the denominator. If it were possible, the process would expose any hidden common factors. 

2) Identify any Holes

We can essentially skip this step; holes are generated when a common factor between the numerator and denominator exists. We would have identified the common factor in the previous step. 

3) Identify any Zeros

Setting the numerator equal to zero allows one to identify the zeros. The numerator of this rational function is not complex by any means, so it is relatively easy to find the zero.

Since we are solving for a zero, the y-coordinate equals zero; thus, there is a zero at \(\left(\frac{5}{3},0\right)\). 

4) Identify any Asymptotes

Of course, there are three types of asymptotes (vertical, horizontal, and oblique), so we need to be sure to take all of them into account, if they exist. 

Setting the denominator equal to zero reveals the vertical asymptote. 

The horizontal asymptotes can be determined by the degree of both the numerator and denominator. In this case, the degree of the numerator and denominator are equal, so you would divide the leading coefficients of the numerator and denominator. 

The horizontal asymptote exists at \(y=\frac{-3}{-5}=\frac{3}{5}\)

For rational functions, it is impossible that an oblique asymptote exists if a horizontal asymptote does, so there is no oblique asymptote. 

5) Plot any Information Determined Previously

We know where a zero exists already (at \(\left(\frac{3}{5},0\right)\)), so we might as well plot it.

Plotting asymptotes are also important; they tell where functions approach, so the function does not cross asymptotes. Be careful, though! For rational functions, a function will never pass a vertical asymptote, but it can pass a horizontal or oblique asymptote. In this case, though, the function will not pass through any asymptotes. 

6) Create a Table of Values

If, after this process, you are still unsure about how a graph behaves, creating a table of values might be your best solution. Be strategic about it, though! Plot points on all sides of vertical asymptotes to better understand the behavior. 

Before starting any problems listed above, you should understand what a similar polygon means. In general, it means that all corresponding angles in two figures are congruent and that all corresponding sides are proportional. Proving corresponding sides proportional requires that you find the ratio of corresponding sides and check for equality.

11) 

In this example, both figures are rectangles, so every angle is a right angle. All right angles are congruent. The next step requires us to compare the ratio of the sides. 

\(\frac{20}{40}\stackrel{?}=\frac{24}{48}\)

Notice what has happened here. For the smaller rectangle, the leftward fraction represents the ratio of the shorter side of the smaller rectangle to the longer side. The ratio is the same for the larger rectangle. Both fractions can be simplified.  

\(2=2\)

The ratios are the same, so both rectangles are similar polygons.

12) 

For this problem, simply employ the same strategy as before. The figures are rectangles, so it is already established that the angles are congruent. Now, let's compare the ratios of corresponding sides. 

\(\frac{8}{16}\stackrel{?}= \frac{12}{30}\\ \frac{1}{2}\neq \frac{2}{5}\)

After simplifying both fractions to simplest terms, it becomes clear that the ratio of corresponding side lengths is not proportional. By definition, corresponding side lengths must be proportional in order for two figures to be considered similar. Since this is not the case, these rectangles are not similar. 

13) 

The diagram is not the prettiest work because I cannot figure out how to show congruent angles on this program. I used slipshod cut-off semicircles. 

In the diagram above, \(\angle A\cong \angle X\text{ and }\angle B\cong \angle Y\text{ and }\angle C\cong\angle Z\). Also, \(\frac{2}{4}=\frac{3}{6}=\frac{4}{8}\), so \(\triangle ABC\sim\triangle XYZ\).

14) The beauty of a similarity (or congruence) statement is that it reveals more information than it may first appear. We should not forget what similarity means! Let's look at the individual choices. 

\(m\angle C=m\angle Z\) is true because both letters appear in the same position in the similarity statement \(\triangle AB\textcolor{red}{C}\sim\triangle XY\textcolor{red}{Z}\). Corresponding angles are always congruent in similar triangles. 

\(\frac{\textcolor{red}{AB}}{\textcolor{red}{XY}}=\frac{\textcolor{blue}{YZ}}{\textcolor{blue}{BC}}\) is not true! Notice that \(\triangle \textcolor{red}{AB}C \sim \triangle \textcolor{red}{XY}Z\text{ and }\triangle A\textcolor{blue}{BC}\sim\triangle X\textcolor{blue}{YZ}\). Remember that both ratios must compare a pair of corresponding sides. In the leftward statement, it compares a side of the smaller triangle to a corresponding side of the larger triangle. In the rightward statement, the ratio compares the length of the larger triangle to the shorter one. Notice how the similarity statement shows this discrepancy; notice that the letters are not in the same order.

\(AB\cong XY\) is not true. Yes, it lines up with the similarity statement \(\triangle \textcolor{red}{AB}C \sim \triangle \textcolor{red}{XY}Z\), but corresponding sides are proportional--not congruent. 

\(\frac{\textcolor{red}{BC}}{\textcolor{red}{YZ}}=\frac{\textcolor{blue}{AC}}{\textcolor{blue}{XZ}}\) is true because it matches perfectly with the similarity statement. Notice that \(\triangle A\textcolor{red}{BC}\sim\triangle X\textcolor{red}{YZ}\) and \(\triangle \textcolor{blue}{A}B\textcolor{blue}{C}\sim\triangle \textcolor{blue}{X}Y\textcolor{blue}{Z}\).

15)

6-8-10 and 9-12-15 are a special set of numbers known as Pythagorean triples, which are positive-integer side lengths of a right triangle. Since both side lengths have this property, we can conclude that \(m\angle B=m\angle F= 90^\circ\). An accepted method of proving triangles similar is called Side-Angle-Side Similarity Theorem, which states that the one pair of sides is proportional, and the included angle is congruent. 

\(\frac{6}{8}\stackrel{?}=\frac{9}{12}\\ \frac{3}{4}=\frac{3}{4}\)

The side lengths are proportional, so we can conclude, by the Side-Angle-Side Triangle Similarity Theorem, that \(\triangle ABC\sim\triangle DFE\).

I know that the difference between postulates and theorems can perplex some of the brightest minds. I have noticed that these terms are often used interchangeably, so the confusion is understandable. The following tidbit may help you make an informed decision when dealing with the said crisis. 

A theorem is a generalized mathematical statement provable via other established truths. 

Some examples include the Triangle Sum Theorem, Pythagoras' Theorem, and Vertical Angles Theorem

A postulate is a generalized mathematical statement assumed as truth. 

Some examples include Segment Addition Postulate (If C is between A and B, then \(AC+CB=AB\) ) and the Ruler Postulate (The points on a line can be placed in a one-to-one correspondence with the real numbers)

 In this case, Angle-Angle Similarity Theorem is the appropriate name because the statement can be proved. 

I guess I can attempt, to my best ability, to explain other confusing terms.

A corollary is a generalized mathematical statement heavily derived from other established theorems. 

Some examples include the Corollary of the Triangle Sum Theorem (The acute angles in a right triangle are complementary) and the Corollary of the Isosceles Triangle Theorem (If a triangle is equilateral, then the triangle is equiangular). 

A proposition is a mathematical result, often marked as less important

Some examples include that pi is irrational and 2 is the only even prime number.

A conjecture is an unproved mathematical statement but is believed to be true

An individual with a basic understanding of math can understand the Collatz Conjecture.

I am willing to share another method for the eighth question

#8)

The method I default to uses the following conversion. I will create another table:
 

There is only one unknown here! We only need to find the number of grams that equals the number of moles of Cl₂. There are 35.45 grams of Cl per mole, according to the trusty periodic table. Cl₂ has double the number of molecules as Cl, so  \(1\text{molCl}_2=35.45g*2=70.9g\). In case you are unaware, "STP" is shorthand for standard temperature and pressure.