Master Algebra 1 Word Problems with our Comprehensive Practice

Prove that sinx is not a rational function using the fact that it is not of the form p(x)/q(x) where p and q are polynomials. Then, by using the above proof, prove that sinx does not satisfy a "quadratic equation" of the form:
$$(\sin <!--\; \; -->x{)}^2{f}_2(x)+(\sin <!--\; \; -->x)f_1(x)+f_0(x)=0,$$
where $f_0,f_1,f_2$ are rational functions.
I know that a rational function cannot be zero at infinitely many points unless it is 0 everywhere, but how does one use this information to formulate p(x)/q(x) argument? If anybody could please help.

Write the first six terms of the sequence ${\displaystyle a_n=2n}$

$f:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ be function satisfying $f(x+y)=f(x).f(y)$ for all $x,y\in <!--\; \in \; -->\mathbb{R}$ and $li{m}_{x\to <!--\; \to \; -->0}f(x)=1$ then show that $f(rx)=(f(x){)}^r$ for all $r\in <!--\; \in \; -->\mathbb{Q}$

Find the slope of any line perpendicular to the line passing through (6,12) and (5,2)

Write an equation of a line through (1, -3) and perpendicular to y=1/2x +4

Write the first five terms of the geometric sequence ${\displaystyle a_1=2,r=\frac{x}{4}}$

Given $p\in <!--\; \in \; -->{\mathbb{N}}^{+},p>1$, find irrational numbers $r$ such that the set of all numbers of the form $\{r\cdot <!--\; \cdot \; -->p^n\}$, where $n\in <!--\; \in \; -->\mathbb{N}$ and $\{x\}:=x-<!--\; -\; -->[x]$, is dense in $(0,1)$.

Write ${\displaystyle y+4=\frac{-2}{3}(x+6)}$ in slope intercept form

Prove ${\displaystyle (a\oplus b)\oplus (~a\&~b)=1}$

How to find quadratic function in vertex form from two points?
A WNBA player takes a three-point shot 22 feet away from the basket, The ball reaches its highest point, 16 feet above the floor, when it is 12 feet (horizontal distance) away from the player. As the ball approaches the basket, it is 13.6 feet above the floor when it is 18 feet away from the player.
a. Write a function in vertex form to model the shot of the basketball. (Let x be the horizontal distance from the player and y be the height of the basketball above the floor).
From today's lesson on quadratic formulas in the vertex form, the vertex form is $y=a(x-<!--\; -\; -->h{)}^2+k$, where a is the growth factor, h is the line of symmetry (x-coordinate of vertex), and k is the y-coordinate of the vertex.
From the question, I can get that the vertex is (12,16), so h is 12 and k is 16. So that means the equation is $y=?(x-<!--\; -\; -->12{)}^2+16$. Another point is (18,13.6). I might be able to infer that another point is (22,0), but it wouldn't help much. a is missing.
There is no more information given. My teacher said that many of us would most likely come back tomorrow begging her for answers, so maybe the answer is not purely mathematical, but requires thinking. How do I solve this (for a)?

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$f(x)=9x{e}^x$
Increasing:
a) $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},-<!--\; -\; -->1)$
b) $(-<!--\; -\; -->1,\mathrm{\infty <!--\; \infty \; -->})$
c) $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},0)$
d) $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->})$
e) $(0,\mathrm{\infty <!--\; \infty \; -->})$
f) no interval

Given the first term and the common dimerence er an arithmetic sequence how do you find the 52nd term and the explicit formula: ${\displaystyle a_1=12,d=-20}$?

Find the equation of the line perpendicular to ${\displaystyle y=\frac{21}{4}x}$ that passes through (−3,5)

Find the slope of a line that is perpendicular to the graph of : y= -1/2x + 4

Solve:
$$(8x+1)(x^2+x+9)-<!--\; -\; -->(x^3-<!--\; -\; -->8)$$

Prove that the system of differential equations
$$\begin{gathered} \stackrel{\dot{}<!--\; \dot{}\; -->}{x}=y \\ \stackrel{\dot{}<!--\; \dot{}\; -->}{y}=1+x^2-<!--\; -\; -->(1-<!--\; -\; -->x)y \end{gathered}$$
doesn't contain periodic solutions.

Find a point of discontinuity, if any
$$f(x)=\frac{2x^2+2x-<!--\; -\; -->60}{x^2-<!--\; -\; -->x-<!--\; -\; -->20}$$

Fingernails grow about 1.5 inches per year. How long would it take to grow nails 37 inches long?

Find the next three terms in the geometric sequence 9, -3, 1, , ...

Tell whether the sequence ${\displaystyle 1,1\frac{1}{6},1\frac{1}{3},1\frac{1}{2}}$ is arithmetic and if it is, what is the common difference