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
3
votes
1 answer

Express implicit equation explicitly

The implicit expression $(b-a)=(a+b)^3$ looks like it could be written explicitly for $a$ as a function of $b$. The only region of interest is for $a,b>0$ Here is what the plot looks like:
Gus
  • 1,491
3
votes
2 answers

Help with finding the real zeros of a polynomial

$$P(x)=x^4-6x^3+4x^2+15x+4$$ Steps I took: Possible zeros are: $$(x+1)(x-1)(x+2)(x-2)(x+4)(x-4)$$ Used synthetic division to find which zero is an actual zero: (I apologize for the graphical representation, I don't know how else to represent it on…
Cherry_Developer
  • 4,239
3
votes
3 answers

How to show this equals 1 without "calculations"

We have $$ \sqrt[3]{2 +\sqrt{5}} + \sqrt[3]{2-\sqrt{5}} = 1 $$ Is there any way we can get this results through algebraic manipulations rather than just plugging it into a calculator? Of course, $(2 +\sqrt{5}) + (2-\sqrt{5}) = 4 $, maybe this can in…
tony
  • 33
3
votes
4 answers

What does the Fundamental Theorem of Algebra say about the number of complex zeros of a polynomial function?

I was watching the Khan Academy video on the Fundamental Theorem of Algebra when I got confused by something that Sal Khan states. From what I understand, the Theorem says that the complex zeros of a polynomial function always come in pairs because…
Cherry_Developer
  • 4,239
3
votes
1 answer

How to isolate x in this case?

We have the following equations: $x^3 + px + q = 0$ $x = u + v $ $p=-3uv$ Where $q$ and $p$ are known real numbers and $x$ is an unknown real number. We want to find an equation for $x$ by first writing one for $u$ and $v$. So we get: $$ (u+v^3) -…
Preet Singh
  • 73
3
votes
2 answers

Thinking about mixtures and percentages

The question in the book was "What percentage of a $35\%$ solution of alcohol in water should be replaced by pure alcohol to give a solution containing 75% alcohol? Assume that the original solution contains 100 units." Here's how I answered…
Juan de la John
  • 135
3
votes
1 answer

Derivative of Logarithmic Rational Function

The function: $$f(x) = \frac{1}{x(\ln x)^2}$$ may be rewritten as: $$f(x) = (x^{-1})(\ln x)^{-2}$$ We begin by using the product rule for differentiaton: $$[f(x)g(x)]' = (-x^{-2})(\ln x)^2 + (x^{-1})(-2)(\ln x)^3)(x^{-1})$$ and is simplified…
cryptomath
  • 385
3
votes
2 answers

Comparing the size of square roots

How to compare the size of following numbers without using the calculator? $a=\sqrt{2}+\sqrt{6}+\sqrt{7},$ $b=\sqrt{3}+\sqrt{4}+\sqrt{8},$ $c=\sqrt{5}+\sqrt{5}+\sqrt{5}$
guest
  • 33
3
votes
1 answer

What is FOIL and how is it done?

My Algebra teacher was explaining to the class about creating a trinomial using FOIL. What is it, and how is it done?
Scribblenautical
  • 195
3
votes
4 answers

Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$

Solve the following equation over the reals:$$ x^3 + 1 = 2\sqrt[3]{{2x - 1}} $$ I noticed that 1 is a trivial solution, then I tried raising the equation to the 3rd, then dividing the polynomial by $(x-1)$.. But I can't see the solution, how do I…
Michael
  • 39
3
votes
2 answers

Finding the domains in which $f(x)>g(x)$

Consider the functions $f(x)=\sqrt{5+4x-x^2}$ and $g(x)=|x-3|+2$. In what domain of $x$ the function $f(x)$ lie above the function $g(x)$, i.e. $f(x)>g(x)$? I think I should subtract the function $f(x)$ from $g(x)$, find where the new function…
Infinity
  • 161
3
votes
4 answers

A summation that starts from $-\infty$

I need to evaluate $\large\sum_{i=-\infty}^{n} 2^i$. This is what I tried: $$S = 2^{-\infty} + \color{red}{2^{-\infty + 1} + 2^{-\infty + 2} + ... + 2^{n-1} + 2^{n}}$$ $$2S = \color{red}{2^{-\infty + 1} + 2^{-\infty + 2} + 2^{-\infty + 3} + ... +…
David G
  • 4,277
3
votes
1 answer

Calculate domain of $f(x) = \sqrt{\frac{1}{x-4} + 1 }^2 -9$

I ran this through the MathWorld domain/range widget, and got $x \leq 3$ & x > 4. I understand that x=4 is undefined, therefore the domain of x includes x > 4, but I cannot for the life of me figure out how to determine the $x \leq 3$. When I set up…
Diogenes
  • 41
3
votes
2 answers

should rate of change be negative

Say I have a spherical snowball. I want it's average rate of change of surface area as radius goes from 25cm to 20cm. I did the calculation. $f(r)=4*\pi*r^2$. That's the formula of surface area of sphere So I did $\dfrac{f(25)-f(20)}{25-20}$ Which…
kirish
  • 31
3
votes
3 answers

How do I find the equation of a parabola given the max, and two points?

I was given the points (2, -1) and (10,-1) and also a max of 4. How would I go about finding the equation of the parabola given this info?
Max
  • 121
1 2 3
…
99 100