# 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.

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### Disaggregating algebra formula

I have been trying to come up with a way to calculate the individual contribution of each row i. However due to the non-linear relationship implied by the power to 0.5 it is not as straight forward as ...
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### Application of rational functions [closed]

can anybody help with this problem I found on my textbook (no this is not my homework I am just doing some training), I believe is very hard and even the answer provided I the book uses approximation: ...
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### Can the graph of a rational function cross oblique asymptotes

I have a follow up question from my previous one: I was told that horizontal asymptotes can be crossed by the graphs of rational functions (and I discovered that the vertical ones can't be crossed). ...
1 vote
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### For what value of $k$ will $kx^2+x+k=0$ have two real solutions?

The answer key says $-1/2 < k < 1/2$, yet when I tried $k=0$ in the equation, I received only one solution of $x=0$. However, the discriminant with $k =0$, is still greater than $0$. What is ...
1 vote
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### Is there a way to prove two polynomial $u(x)$ and $v(x)$ are always equal or not for any $A$? [closed]

Is for any $A \geq 0$ in rational field, $u(x)=v(x)$ always? Where, $u(x)= x^6+2x^3+1$ and $v(x)= (1+x^2-2Ax)^3$
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### Understanding why a fraction is the same as a division problem [duplicate]

I am trying to understand why fractions are another way to write division. I haven't seen a satisfying explanation for this yet. I know that division when we say for example a divided by b is like ...
1 vote
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### Intuition behind the sum $\sum_{n=1}^{\infty} \frac {n-1}{n!}=1$? [duplicate]

I've been fiddling around with some sums, and got this result: $$\sum_{n=1}^{\infty} \frac {n-1}{n!}=1$$ I've managed to prove this using the Taylor series of e^x, but I'm looking for some Intuition/...
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### I wonder where the author used (I) in the above proof. ("Linear Algebra" by Ichiro Satake.)

I am reading "Linear Algebra" (in Japanese) by Ichiro Satake. Theorem 2: The necessary and sufficient condition for $m$ $n$-dimensional vectors $a_j = (a_{ij})$ ($1 \leq j \leq m$) to be ...
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### View minus sign as operator or part of the number? How to differentiate? [closed]

I came across this problem,looking at the distributive law "a*(b+c) = ab+ac" / "a*(b-c) = ab-ac". Lets say we have the following term: -4 * (2 - 4) What would you say is c? Is c -4 ...
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### Proof of Unique Factorisation of Polynomials over $\mathbb C$ by Identity Principle

Proposition: Any polynomial $p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ can be expressed uniquely as $$p(x) = a_n(x-r_1)(x-r_2)\cdots(x-r_n),$$ where $r_1, r_2,\ldots, r_n$ (not necessarily ...
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### I don't understand how difference of vectors work {HOMEWORK} [duplicate]

So in the picture we have vectors u and v. Our goal is to find $v−u$ From what I know, the subtraction of vectors is just reversing the direction of the $2^{nd}$ vector & then finding the ...
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### Inequalities and averages

Dikshant writes down $2 k+1$ positive integers in a list where $k$ is a positive integer. The integers are not necessarily all distinct, but there are at least three distinct integers in the list. The ...
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### What is the Maximum Theoretical Angle a Grand Piano Could be Held At?

Out of curiosity, I wondered why grand pianos have their stand at the length and position that they are made at. I never could find an answer so I decided to try to solve for the maximum angle (B) the ...
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### Summation involving the closest integer to $\sqrt n$ [closed]

Let $f(n)$ be the integer closest to $\sqrt n$. Evaluate $$\sum_{n=1}^\infty\frac{\left(\frac32\right)^{f(n)}+\left(\frac32\right)^{-f(n)}}{\left(\frac32\right)^n}$$ In this question, I was able to ...
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### Solution-verification: Solve $3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr)$

the problem Solve in the set of real numbers the following equation $$3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr),$$ where $\lfloor x\rfloor$ and $\{x\}$ are the whole part and the ...
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### Find the integers solutions of the equation $x^4+4y^4=3796$ [closed]

the problem Find the integers solutions of the equation $x^4+4y^4=3796$ my idea First thing that came into my mind is that $x^4=4(949-y^4)\Rightarrow 4|x^4 \Rightarrow 2|x$ which means that x is ...
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### Show that $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$

Let the real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=3$. Show that: $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$. My idea: First of all, I thought ...
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### Faster way to find self-intersections of the curve parameterized by $(-4t^3-6t^2,-3t^4-4t^3)$

Given is a curve $K$ with $K(t)=\begin{pmatrix}f(t)\\g(t) \end{pmatrix}=\begin{pmatrix}-4t^{3}-6t^{2}\\-3t^{4}-4t^{3} \end{pmatrix}$ and $-1.5 \leq t \leq 0.5$. I want to find the intersection of the ...
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