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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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Factoring question from March $2013$ AMATYC exam

For how many pairs of positive integers $(n, \space m)$ with $n, \space m < 100$ are both of the polynomials $x^2 + mx + n$ and $x^2 + mx - n$ factorable over the integers? I have found four solutions: $$x^2 + 10x + 24$$ $$x^2 + 20x + 96$$ $$x^2 +…
Kara
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Find the numbers satisfying $x+y=19$ and $x^3+y^3=2071$

The sum of the cubes of two numbers is $2071$, while the sum of the two numbers themselves is $19$. Find the two numbers. I've been working hard to solve this problem and I need someone to tell me how to solve it, it perplexes me. I know the answer…
user123940
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Solve for variable

$$0 = \frac2{r-1}-\frac3{r+4}+\frac1{r+5}$$ So to my understanding I could give them all the same denominator by multiplying their denominators with each others denominators and numerators. Or could I just flip them all around? But that would mean…
John
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Determine the cube roots of -8 in polar form

Exam time tomorrow and I am not entirely sure if I am doing this right. I first write -8 as a complex number $z^3 = -8 = -8 \times 0i$ Calculate the modulus of z $|z| = \sqrt{-8^2} = 8$ Get the arg of z $tan^{-1} = \frac{0}{-8} = 0 = \pi$ Write the…
Leon
  • 751
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For what $(m, n)$, does $1+x+x^2 +\dots+x^m | 1 + x^n + x^{2n}+\dots+x^{mn}$?

For what $(m, n)$, does $1+x+x^2 +\dots+x^m | 1 + x^n + x^{2n}+\dots+x^{mn}$? Well, $$\sum_{i = 0}^{m} x^i = \frac{x^{m+1} - 1}{x - 1}$$ and, $$\sum_{i = 0}^m x^{in} = \frac{x^{n(m+1)} - 1}{x-1}$$ Notice that $x^{m+1} - 1|(x^{m+1})^n - 1$,…
Gerard
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$ \frac1{bc-a^2} + \frac1{ca-b^2}+\frac1{ab-c^2}=0$ implies that $ \frac a{(bc-a^2)^2} + \frac b{(ca-b^2)^2}+\frac c{(ab-c^2)^2}=0$

$a ,b , c$ are real numbers such that $ \dfrac1{bc-a^2} + \dfrac1{ca-b^2}+\dfrac1{ab-c^2}=0$ , then how do we prove (without routine laborious manipulation) that $\dfrac a{(bc-a^2)^2} + \dfrac b{(ca-b^2)^2}+\dfrac c{(ab-c^2)^2}=0$
Souvik Dey
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Child's homework equation related to algebra

My grandson's homework.... There are 23 fish. Guppies (G) are 3 more than Z-fish (Z). There are 2 times the Z-fish as Goldfish (GF). How many of each? .... I can see there are 8 Z-fish (Z), 11 Guppies (G) & 4 Goldfish (GF).... HOWEVER, When I try…
Cynthia
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$a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$

How can we show that the assumption $a,b,c>0$ and $a+b+c=1$ implies $$\sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}~?$$
Eisen
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$f(x,y) = \sqrt{x^2+(y-1)^2}+\sqrt{(x-3)^2+(y-4)^2}-\sqrt{x^2+y^2}-\sqrt{(x-1)^2+y^2}\;\;,x,y\in \mathbb{R}$.

Let $f(x,y) = \sqrt{x^2+(y-1)^2}+\sqrt{(x-3)^2+(y-4)^2}-\sqrt{x^2+y^2}-\sqrt{(x-1)^2+y^2}\;\;,x,y\in \mathbb{R}$. Then Max. of $f(x,y)$. $\underline{\bf{My\;Try}}::$ We can convert into Complex no. form ... Let $z=x+iy$, Then $f(z) =…
juantheron
  • 53,015
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How many Integer values of $n$ are possible for $n^2+25n+19$ to be a perfect square.

[1] How many Integer values of $n$ are possible for $n^2+25n+19$ to be a perfect square. [2] How many Integer values of $n$ are possible for $n^2-19n+99$ to be a perfect square. $\underline{\bf{My\;Try}}::$ for first one , Let $k^2 = n^2+25n+99$,…
juantheron
  • 53,015
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An awful factorisation question.

If $a+b+c = 0$ show that $$(2a-b)^3 + (2b-c)^3 + (2c-a)^3 = 3 (2a-b)(2b-c)(2c-a)$$ I have tried substituting the values but it gets too complicated. Can anyone please help with the method? I have been trying for 30 minutes. Thanks!
Maga Kriv
  • 51
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Is this operation legal?

Is this operation allowed? Going from this: $\left ( \frac{x^{2}+6}{x^{2}-4} \right )^{2}= \left ( \frac{5x}{4-x^{2}} \right )^{2}$ To this: $\left ( \frac{\left (x^{2}+6 \right )\left ( 4-x^{2} \right )}{\left (x^{2}-4 \right )5x} \right )^{2}=…
lazyCrab
  • 227
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Finding the intersection of two functions

Find the point of intersection between $y=\sqrt{x}$ and $y=\dfrac{x^2}{8}$. I know that I have to set them equal to each other and solve for zero. The problem I have is that these terms are nice and will not factor nicely. This is as far as I got…
Kot
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Why can't I take the square root of both sides when finding the roots of a quadratic equation?

Is $x^2 - x - 12 = 0$ equivalent to $x = \sqrt{x + 12}$? I started with $x^2 - x - 12 = 0$ and made the following changes: $x^2 - x - 12 = 0$ $x^2 = x + 12$ $x = \sqrt{x + 12}$ From here I can eyeball it and see that x = 4 and x = -3 are…
dumb question
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roots of unity: Finding $ \sum_{i=1}^{2n}\frac{r_i^2}{r_i+1} $, where the $r_i$ are the roots of $x^{2n} + x^{2n-1} + x^{2n-2} + \cdots + x + 1$

A high school polynomial root question: Let $W(x)=x^{2n} + x^{2n-1} + x^{2n-2} + \cdots + x + 1$. Let the root of $W(x)$ be $r_1, r_2, ... r_{2n}$. Find $$ \sum_{i=1}^{2n}\frac{r_i^2}{r_i+1} $$ in terms of $n$. What I found: $r_i^{2n+1}=1$…
xwa130
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