Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Rationalizing the denominator with 3 roots

Well, I can't find the example on how to solve this. If I multiply $$ \dfrac{2}{\sqrt[3]{9}+\sqrt[3]{15}+\sqrt[3]{25}} $$ with $$ \dfrac{\sqrt[3]{9}-\sqrt[3]{15}+\sqrt[3]{25}}{\sqrt[3]{9}-\sqrt[3]{15}+\sqrt[3]{25}} $$ or similar, it just gets even…
Guest114
  • 33
3
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3 answers

Proving that a $f:\mathbb{R} \to (-1,1); x \mapsto \frac {x}{\sqrt{1+x^2}}$ is bijective

I am trying to prove that a function is bijective and I really am not sure how to go about it. I know that I must show that the function is both injective and surjective for it to be bijective. The function that I am trying to prove is bijective…
Aaron
  • 31
3
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3 answers

Is every real root also a complex root? (Complex zeros and Fundamental Theorem of Algebra)

Stupid question about this topic: I don't understand the rule that says "A poly with degree N will have N complex roots". eg: $P(X)=x^5 + x^3 - 1$ is a 5th degree polynomial function, so P(x) has exactly 5 complex zeros. Sorry, but how do you…
JackOfAll
  • 4,701
3
votes
1 answer

Calculation of integers $b,c,d,e,f,g$ such that $\frac{5}{7} = \frac{b}{2!}+\frac{c}{3!}+\frac{d}{4!}+\frac{e}{5!}+\frac{f}{6!}+\frac{g}{7!}$

There are unique integers $b,c,d,e,f,g$ such that $\displaystyle \frac{5}{7} = \frac{b}{2!}+\frac{c}{3!}+\frac{d}{4!}+\frac{e}{5!}+\frac{f}{6!}+\frac{g}{7!}$ Where $0\leq b,c,d,e,f,g
juantheron
  • 53,015
3
votes
3 answers

How do you find the cube roots of just a number?

I was told to write the $3$ cube roots of $27$ in rectangular form how is this to be done? I know how to do cube roots on something like $5+0i$ in polar form, but how do you do it with just a number?
chromedude
  • 229
3
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0 answers

Find all real roots of $x^5+10x^3+20x - 4=0$

Find all real roots of the following fifth-degree equation $$x^5+10x^3+20x-4=0.$$ In fact, the only one root is $\sqrt[5]8-\sqrt[5]4$. How to get that?
ziang chen
  • 7,771
3
votes
1 answer

Find two unknowns that have a certain ratio.

I need to find a value for L and S where: P = 20 L = 2S + P L + S + 3P = 1160 It's for deciding the column width in a website layout, which can change according the available screen size: I tried giving L a random value that sounded about right…
Bart van Heukelom
  • 183
3
votes
2 answers

A theory of equation question from my exam paper

Consider The equation $x^3+3x^2+3x+3=0$ Then the sum of it's non-real roots is A) is equal to $0$ B)lies in $0$ and $1$ C)lies in $-1$ and $0$ D)Greter that $1$ Which one is correct , plz explain briefly;
tensor
  • 134
3
votes
5 answers

Determine limit of |a+b|

This is a simple problem I am having a bit of trouble with. I am not sure where this leads. Given that $\vec a = \begin{pmatrix}4\\-3\end{pmatrix}$ and $|\vec b|$ = 3, determine the limits between which $|\vec a + \vec b|$ must lie. Let, $\vec b =…
mathguy80
  • 1,321
3
votes
2 answers

Simple condition in a Math expression

Is there any simple math expression that returns $-1$ if $x > a$ or $1$ if $x < a$?
Alon Gubkin
  • 634
3
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3 answers

$x^4+4$ is composite for $x>1$

$x^4+4$ is composite for $x>1$ I know the Sophie Germain indentity and the get the factorization $$x^4+4 = (x^2+2-2x)(x^2+2+2x)$$ But I am stuck here. I cannot see any general factor here.
kuch nahi
  • 6,789
3
votes
2 answers

Roots of a Quadratic Problem

I'm struggling with this problem and was hoping I could get some advice. Here is the problem: Let a and b be the roots of the quadratic equation $x^2−x−1/27=0$. Without calculating the a and b show that $a^{1/3}+b^{1/3}$ is a root of the equation…
Hummus
  • 563
3
votes
2 answers

Can anyone explain to me this square root? step by step?

$$\begin{align} v(p_1, p_2, w) & = \sqrt{\frac w{p_1^2\left(\frac1{p_1}+\frac1{p_2}\right)}} + \sqrt{\frac w{p_2^2\left(\frac1{p_1}+\frac1{p_2}\right)}} \\ & = \sqrt{\frac…
Maximilian1988
  • 1,323
3
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0 answers

Inverse of $f(x) = x + e^{2x}$

How do you find the inverse of $f(x) = x + e^{2x}$ ? I started by trying to find the inverse by replacing $f(x)$ with $y$, switching $x$ and $y$, and solving for $y$. $$y = x + e^{2x} \implies x = y + e^{2y}$$ I took the natural log of both sides…
user102658
  • 31
3
votes
0 answers

Transforming a system of equations (line or any curve)?

Suppose $S=\{v:Av=b\}$ is the solution set to a system of $m$ equations in $n$ variables. How do I write a system of equations that give the solution set $\{Pv:v\in S\}$, where $P$ is some $p\times n$ matrix? The new system of equations would be in…
user13727
  • 31
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