# 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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### necessary and sufficient conditions?

What are the conditions under which a real-valued function of real variables $f(x,(y_1,...y_n))$can be written as $\sum_{i=1}^{N} g_i(x,y_i)$
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### Why is considering only quadratic in one of the variables of a two variable quadratic sufficient for calculating roots

Find the $positive$ integral solutions to $7x^2-2xy+3y^2-27=0$ My solution: Assuming the quadratic in $x$ , if we assume one root to be integral , the other has to be rational (as y must be an ...
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### All integer solutions of $x^3-y^3=2020$.

Find all integer pairs $(x,y)$ satisfying $x^3-y^3=2020$. First, $x^3-y^3=(x-y)(x^2+xy+y^2)=2020$ and $2020=2^2\cdot 5 \cdot 101$. But what next? Can it be worked out by using modulo? Or how? Any ...
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### Prove that $\exists !c \in \mathbb{R} \exists ! x \in \mathbb{R} (x^2 + 3x + c = 0)$

This is an exercise from Velleman's "How To Prove It". I am struggling with how to finish the final part of the uniqueness proof, so any hints would be appreciated! a. Prove that there is a ...
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### Finding Maclaurin series f(x)

Can anyone please help me with finding Maclaurin series for this $$f(x) = x^3 \tan^{-1}(2x); \quad |x|<\frac12$$ https://i.stack.imgur.com/bUhxk.jpg
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### Can a real 2 by 2 matrix have one eigenvalue with geometric multiplicity 2?

Given the a real matrix $A=\begin{bmatrix} a & b \\ c& d\end{bmatrix}$, we assume that it has only one real eigenvalue $\lambda$. I am wondering if it is possible that the eigenvalue $\lambda$ ...
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### Spivak Calculus chapter 1 problem 13 proof critique

Question paraphrased: Prove that the maximum between two numbers $x$ and $y$ is given by: $$\max(x,y)=\frac{x+y+|y-x|}{2}$$ Proof: Let $x$ and $y$ be two arbitrary numbers. Then, one and only one ...
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### Can I use this rule in Spivak Calculus chapter 1?

In proof question no. 12 (iv), I used $|\frac{x}{y}| = -\frac{x}{y}$ if $x>0$ and $y<0$. However I am not sure if Spivak has defined that $\frac{1}{-y} = -\frac{1}{y}$. Should I take this as a ...
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### How does the exponent behave when changing numerator and denominator?

I have seen this equation: $$\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$$ As you can see the numerator switched with the denominator and I wonder how. I know ...
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### does p = np in the hard set of problems [closed]

just got asked this in my year 6 homework questions. Apparently it is really easy according to my teacher. Any help would be appreciated. thnx
find function f(x) such that the following holds \begin{align} e^{\sum_{i=1}^{n} f(x_i)} = \sum_{i=1}^{n} x_i \end{align} where $0<x_i<1 \quad \forall I$ and n is an integer n>1. or, if the ...