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.
47,622
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Positive, Real Roots of Bivariate Polynomial
I have a question regarding lemma 3.1 in this paper. The lemma in question is as follows
Consider the function $f(x, \lambda) = ax^3 + bx^2 + cx + d$ where $a > 0$ is
fixed but for which the ...
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Handling of algebra in differential calculus
3Blue1Brown"Essence of calculus" series called "Derivative formulas through geometry"- 3rd episode of chapter 3
I have considered the area gained to be $dx*(\frac{1}{x}-d(\frac{1}{...
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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" 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 linearly ...
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View minus sign as operator or part of the number? How to differentiate?
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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Solve $|x|>|x-1|$
Solve $|x|>|x-1|$
$\dfrac{|x|}{|x-1|}>1 \Leftrightarrow \left| \dfrac{x}{x-1} \right| >1$
$\dfrac{x}{x-1} > 1 \tag{1}$ or $-\dfrac{x}{x-1}>1 \Leftrightarrow \dfrac{x}{x-1}<-1 \tag{...
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Solution of equation with unknown under the integral
I have a problem which I have reduced to solving the following equation for the unknown $r_0$:
$$
1/2 = \int_0^D f(r)p(r,r_0)dr
$$
where $D \in \mathbb{R}$, and $f$ is continuous density function.
$p(...
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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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Precise Definition of Polynomial [duplicate]
Apologies if this question is too trivial. I am having trouble precisely defining polynomials. All of the definitions I have seen say that expressions of the form $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+...
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Solving $ \left|\frac{3x}{7} \right |= 4-x$
I’m trying to solve:
$\displaystyle \left|\frac{3x}{7} \right |= 4-x$
Here’s what I’ve tried:
$\frac{3x}{7} = 4-x$ (checking for intersections)
$x = \frac{14}5$ (this intersection checks out)
...
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AM-GM inequality for non necessary positive numbers
For nonnegative real numbers $x_1,\cdots,x_n$ ($n\geqslant2$), it is well known that :
$$n\prod_{i=1}^nx_i\leqslant\sum_{i=1}^nx_i^n\tag{$\star$}$$since this is equivalent to the AM-GM inequality.
But ...
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Finding a value $n$ such that $\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$ is true.
For what value of $n \in \mathbb{N}$ such that the following inequality is true.
$$\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$$
Where $0<x\le \sqrt[5]{216}$
ATTEMPT:
This is my first time tackling ...
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