Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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1answer
32 views

Quick question about basic proofs (Spivak)

I was taking a look at some of the practice question from the first chapter of Spivak, and I am wanting to just verify if I am on the right track with things. I do not have solutions, so I am just ...
-1
votes
2answers
58 views

How do I find $\frac {x^3}{x(x-3)}$ partial fractions?

How do I find $\frac {x^3}{x(x-3)}$ partial fractions? And in general, when the degree of the numerator is higher than the denominator's? Thanks in advance for your assistance!
2
votes
1answer
82 views

How to find coefficient of $x^{12}$ in the expansion of $(1+x+x^2+x^3+…+x^n)^4$

How to find coefficient of $x^{12}$ in the expansion of $(1+x+x^2+x^3+...+x^n)^4$ I tried this : Since $(1+x+x^2+x^3+...+x^n)$ is in GP its sum will be $(x^{n+1}+1)(x-1)^{-1}$ now ACQ we have to ...
1
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3answers
71 views

$2\sin(\theta + 17) = \dfrac {\cos (\theta +8)}{\cos (\theta + 17)}$

For $0<\theta<360$ $$2\sin(\theta + 17) = \dfrac {\cos (\theta +8)}{\cos (\theta + 17)}$$ $$\Longrightarrow \sin(2\theta + 34)= \sin (82-\theta)$$ since sine is an odd function $$2\theta + ...
1
vote
3answers
34 views

Finding extrema.

Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ . I used the formula for distance between two points in a plane to get: ...
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4answers
53 views

Solving Trig Equation $\cos(2x)=-\sin(2x)$

Proceeding as follows: $$\cos(2x)=-\sin(2x)\Rightarrow \cos \left(2x\right)=-\cos \left(\frac{\pi }{2}-2x\right)$$ How to proceed further? Can I remove the $cos$ from both sides and proceed or no?
0
votes
2answers
86 views

How to factorise $x^4$ equations?

This is my previous question I'm facing a problem to factorise this $64x^4+64x^3-88x^2-51x+39=0$. How to factorise $x^4$ equations?
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3answers
69 views

How can I prove that $f$ doesn't have all real roots $\forall a\in\mathbb{C}$

We have $f=x^4+ax^3+4x^2+1\in\mathbb{C}[x]$ with $x_1,x_2,x_3,x_4\in\mathbb{C}$. We need to prove that $\color\red{\forall a\in\mathbb{C}},f$ doesn't have all real roots. How can I begin to solve ...
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votes
2answers
46 views

The question is x+(x÷4)=60? [closed]

How to solve? I can not solve it . It is very difficult.
2
votes
3answers
473 views

Change from product to sum

We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : ...
1
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2answers
54 views

Tough Polynomial Root Problem

Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either ...
3
votes
3answers
63 views

needs solution of the equation ${(2+{3}^{1/2}})^{x/2}$ + ${(2-{3}^{1/2}})^{x/2}$=$2^x$

$$\left(2+{3}^{1/2}\right)^{x/2} + \left(2-{3}^{1/2}\right)^{x/2} = 2^x.$$ Clearly $x = 2$ is a solution. i need others if there is any. Please help.
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3answers
146 views
+50

A lot of confusion in the “Polynomial Remainder Theorem”?

Lately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynomial ...
2
votes
4answers
124 views

A man died. Let's divide the estate!!! How?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with plain old algebra, which yields the shortest, simplest answers, but other ...
2
votes
2answers
58 views

$xy + yz + zx + 2xyz = 1$ implies $4x+y+z\geq 2$

Let $x,y,z>0$ satisfy $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x+y+z\geq 2$. The condition invites the factoring $(1+x)(1+y)(1+z)+xyz-2=x+y+z$, but having the factor $4$ in the desired inequality ...
3
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3answers
56 views

How to prove that $a^{\log_cb}=b^{\log_ca}$

I've met a question whereby it asked me to show that $a^{\log_cb}=b^{\log_ca}$. I'm okay with the other logarithm questions. But I don't know how to show this question out. Can anyone give some hints ...
0
votes
1answer
60 views

Inverse function $g^{-1}$

The function $g$ is defined by $$g(x)= 3-2x-4x^2, x\in \mathbb{R},x\leq -\frac{1}{4} $$ Find the inverse function $g^{-1}$. Calculate the value of $x$ for which $g(x)=g^{-1}x$. My attempt, ...
5
votes
1answer
109 views

Question about proving basic results of numbers

I have just recently started to work with Calculus by Spivak and I am just wondering some things about the first chapter. ( I am doing this as a method to review my calculus which I have done but only ...
0
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3answers
41 views

How should I set up the equation for this problem?

A library leases its photocopier. One monthly bill was 750 dollars for 12,000 copies. Another month, the bill was $862.50 for 16,500 copies. How much does the library pay for each copy? How ...
1
vote
1answer
57 views

How do I parametrize a circle that's not centered at the origin?

If the circle were centered at the origin, of radius r, then r(cos$\theta$, sin$\theta$) traverses the circle once counterclockwise, for 0 $\le$$\theta$$\le$2$\pi$. What if the circle were centered ...
2
votes
3answers
68 views

Distance between points (think geometrically)

For this problem do I use the distance formula that I would use between two regular points? $d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}$ The distance between points $u$ and $v$ on the $x$-axis is given by ...
2
votes
4answers
37 views

Prove the identity $\frac{1}{\tan (x)(1+\cos( 2x))} = \csc(2x)$

$$\frac{1}{\tan (x)(1+\cos(2x))} = \csc(2x)$$ I really don't know what to do with denominator. Sure, I can use the double angle formula for cosine, and get: $$\frac{1}{\tan(x)(2 - 2\sin^2(x))} = ...
1
vote
1answer
33 views

If log8n=1/2p, log22n=q, and q-p=4, find n [duplicate]

I'm having a hard time finding the value of $a$ in this problem. My teacher was trying to explain to me the process in which to get it but I did not understand him.
0
votes
2answers
48 views

Find $a$ and $b$ such that $f(x) = ax^3 + bx^2 - 28x + 15$ is divisible by $(x+3)$, etc

Find the values of $a$ and $b$ if the polynomial $f(x) = ax^3 + bx^2 - 28x + 15$ is exactly divisible by $(x+3)$ and leaves a remainder of $-60$ when $f(x)$ is divided by $(x-3)$. Use these values ...
2
votes
2answers
43 views

Solve the follwing system of equations for $x, y$ and $z$

$$\frac{y+z}{5}=\frac{z+x}{8}=\frac{x+y}{9}$$ and $$6(x+y+z)=11$$ My teacher told me that I would have to get $3$ different equations to get $x, y$ and $z$. I've tried many methods and I'm confused ...
1
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1answer
39 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
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6answers
60 views

Can anybody help me with this logarithm problem?

If $(\log_4 x)^2= (\log_2 x)(\log_a x)$ (the $4$ is the little number next to log by the way) , find the value of $a$.
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4answers
43 views

Find $f(a)$, $f(a + h)$, and the difference quotient, given $f(x)=7-8x+2x^2$

Question is here: I need help with part c. I tried plugging everything in and simplifying to a point where my final answer was $(-8h+2ah+2h^2)/h$ My work: $(7-8a-8h+2(a+h)^2-7+8a-2a^2)/h$ ...
-4
votes
1answer
49 views

When the logarithm and integral can be commuted? [closed]

When the Logarithm and integral can be commuted?
0
votes
1answer
18 views

Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these.

Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these. I want to compute all the extreme points of the set $P$ (polyhedron) in $\mathbb R^3$ ...
1
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2answers
62 views

Limit of a complicated function.

Find $$\lim\limits_{x \to 2^{-}} \frac{e^{((x+2)\log 4){\frac{[x+1]}{4}}}-16}{ 4^x -16}$$ where $[x]$ denotes the greatest integer function less than or equal to x. ATTEMPT: I tried the following ...
3
votes
2answers
44 views

Can $\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$ be reduced to a convex hull of a subset of these?

How do I see whether $$\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$$ can be reduced to a convex hull of a subset of these vectors? That is, if $D = \{ e_1,e_2,e_3, ...
3
votes
2answers
55 views

$1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$

The inequality $1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$ is true for all $a,b,c\in\mathbb{R}$? I've tried to use the ...
2
votes
1answer
16 views

Binomial expansion in descending power

For example, find, in ascending powers of $x$, the first three terms in the expansion of $(2+5x)^7$. So, $(2+5x)^7=2^7+\binom{7}{1}(2^6)(5x)+\binom{7}{2}(2^5)(5x)^2$. I've no problem to solve this ...
1
vote
2answers
51 views

Show that 10 lines pass through the same point.

Let $A$, $B$, $C$, $D$, and $E$ be five points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to the ...
1
vote
5answers
74 views

Could translate/explain this for me?

I have this problem: $$ 10x^2 - 7x - 12 = 0 $$ And apparently the method to factoring it is to find two numbers whose product is the same as the product of the coefficient of $x^2$ and the constant ...
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2answers
32 views

Distance between orthocenter and circumcenter.

Let $O$ and $H$ be respectively the circumcenter and the orthocenter of triangle $ABC$. Let $a$, $b$ and $c$ denote the side lengths. We are given that $a^2+b^2+c^2=29$ and the circumradius is $R=9$. ...
-4
votes
3answers
57 views

How to prove that: $\log_{{1\over 2}}(3) + \log_3\left({1 \over 2}\right) < -2$ [closed]

Prove that: $$\log_{{1\over 2}}(3) + \log_3\left({1 \over 2}\right) < -2$$ Please help me solve it.
1
vote
1answer
70 views

We're given the function $f(x) = x^2 + bx + c \quad(b,c \in R)$. Value of the function $f(0)$ is?

Given the function $$ f(x) = x^2 + bx + c \quad(b,c \in R) $$ such that $f(f(1)) = f(f(2)) = 0$ and $f(1) \neq f(2)$, what is the value of $f(0)$? Have no basic clue how to do this.
2
votes
5answers
302 views

Triple fractions

I've got this simple assignment, to find out the density for a give sphere with a radius = 2cm and the mass 296g. It seems straightforward, but it all got hairy when i've got to a fraction with three ...
2
votes
4answers
138 views

The number of non-negative real roots of $2^{x}-x-1$ are

The number of non-negative real roots of $2^{x}-x-1$ are $ a.)\ 0\\ b.) \ 1 \\ c.)\ 2 \\ d.)\ 3 \\$ I don't have any clue. I have only learned to solve quadratics and cubic equations , i ...
0
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3answers
37 views

Quick way to solve simple probability questions like these? [duplicate]

Two dice are rolled. What is the chance of rolling at least ONE six? (The answer is $\frac{11}{36}$.) My solutions manual has a very elaborate table drawing method and I don't want to do that. I need ...
0
votes
1answer
27 views

$2(v_{0}^2 - v_{0y}^2)d_{y} = 2v_{0y}d_{x}\sqrt{v_{0}^2-v_{0y}^2} + ad_x^{2}$

I have the following equation: $$2(v_{0}^2 - v_{0y}^2)d_{y} = 2v_{0y}d_{x}\sqrt{v_{0}^2-v_{0y}^2} + ad_x^{2}$$ All variables except $v_{0y}$ are given. How can I find $v_{0y}$ ? Thanks!
3
votes
1answer
31 views

Find the number of equations having real roots.

If both $a$ and $b$ belong to the set $\{1,2,3,4\}$ , then number of equations of the form $ ax^2+bx+1=0$ having real roots is $a.)\ 10\\ \color{green}{b.)\ 7}\\ c.)\ 6\\ d.)\ 12\\ $ To solve ...
0
votes
1answer
97 views

How do I show that $B$ has an infinite number of extreme points, but no faces (sides) of dimension $1$ or $2$?

Let $B = \{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 + z^2 \le 1\}$. How do I show that $B$ has an infinite number of extreme points, but no faces (sides) of dimension $1$ or $2$ ? I've been thinking ...
1
vote
0answers
41 views

$5^{\frac 1 2 -r}=(\frac x 3) ^{2r-1}$, $\pm$ solution?

$$5^{\frac 1 2 -r}=(\frac x 3) ^{2r-1}$$ $$=> \dfrac 1 {5^{r-\frac 1 2}} = (\frac {x^2} {3^2})^{r-\frac 1 2} \,\,\,\,\,\,\,\,\,\,\, (*)$$ $$=> (\dfrac 9 5)^{r-\frac 1 2} =(x^2)^{r-\frac 1 2}$$ ...
3
votes
5answers
234 views

How to solve $2^x+e^x=400$

This should be pretty easy, I know. It involves logs, but then there's this 400. So logs of what? And since $2^x$ and $e^x$ are different things, I can't substitute the values and solve as a second ...
-1
votes
1answer
37 views

For the two questions graph logarithmic functions

For the two questions A) Graph $f(x)$ B) Graph $f^{-1}(x)$ on the same axes. C) Find $f^{-1}(x)$. $f(x)=\log_4(x-3)+2$ $f(x)= e^{x+3}+2$ For number one $x-3$ has an asymptote ...
1
vote
3answers
51 views

Find the remainder when $f(x)$ is divided by $x^2+x-2$

When $f(x)$ is divided by $x-1$ and $x+2$, the remainders are $4$ and $-2$ respectively. Find the remainder when $f(x)$ is divided by $x^2+x-2$. Please help. The answer is $2x+2$. I tried to ...
2
votes
4answers
40 views

Find the value of $x$ such that $(3-\log_3x)\log _{3x}3=1$.

Find the value of $x$ such that $(3-\log_3x)\log _{3x}3=1$. Is there another way to solve other than this attempt? My attempt, $(3-\log_3x)\log _{3x}3=1$ $\frac{\log(3)\left(3-\frac{\log (x)}{\log ...