Articles of algebra precalculus

Do values attached to integers have implicit parentheses?

Given $5x/30x^2$ I was wondering which is the correct equivalent form. According to BEDMAS this expression is equivalent to $5*\cfrac{x}{30}*x^2$ but, intuitively, I believe that it could also look like: $\cfrac{5x}{30x^2}$ I asked this question on MathOverflow (which was “Off-topic” and closed) and was told it was ambiguous. I was wondering what the convention was […]

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have no idea how to get started. Any help would be awesome!

When do we get extraneous roots?

There are only two situations that I am aware of that give rise to extraneous roots, namely, the “square both sides” situation (in order to eliminate a square root symbol), and the “half absolute value expansion” situation (in order to eliminate taking absolute value). An example of the former is $\sqrt{x} = x – 2$, […]

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I’ve found various things that work quite well. For example, when talking about polynomials and rational functions, in particular “zeroes” and “vertical asymptotes”, I introduce them as the same thing, only the asymptotes are “points at infinity”. This (projective plane) model helps the […]

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren’t the graphs the same?

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$’s will work but what else do I do? I think you do $4x+1<-3x$. Is this correct?

Is this a valid partial fraction decomposition?

Write $\dfrac{4x+1}{x^2 – x – 2}$ using partial fractions. $$ \frac{4x+1}{x^2 – x – 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$ $$4x+1 = A(x-2)+B(x+1)$$ $$x=2 \Rightarrow 4 \cdot2 + 1 = A(0) + B(3) \Rightarrow B = 3$$ $$x = -1 \Rightarrow 4(-1) +1 = A(-3)+ B(0) \Rightarrow A = 1$$ Thus, $$\frac{4x+1}{x^2-x-2} […]

Can sum of a rational number and its reciprocal be an integer?

Can sum of a rational number and its reciprocal be an integer? My brother asked me this question and I was unable to answer it. The only trivial solutions which I could think of are $1$ and $-1$. As to what I tried, I am afraid not much. I have never tried to solve such […]

Baby Rudin: Chapter 1, Problem 6{d}. How to complete this proof?

Fix $b>1$. Problem 6(a): Let $m$, $n$, $p$, $q$ be integers such that $n>0$, $q>0$, and $r = m/n = p/q$. Then I’ve managed to prove that $$b^{m/n} = b^{p/q}.$$ So we can reasonably define $b^r$ as $$b^r \colon = \sqrt[n]{b^m}.$$ From this definition, we can Problem 6(b) prove that $$ b^{r+s} = b^r \cdot […]

Solutions to exp(x) + x = 2

I am extremely new to mathematics, and I don’t have much training except for the basics so please excuse my rather basic question. The question itself: If I have the relationship $e^x + x – 2 = 0$; and $k$ is the number of solutions in $[0,1]$ and $n$ is the number of solutions not […]