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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
4
votes
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Counting the number of rotations of a cube.

Take a cube $C$ $[-1,1]^3\subset\Bbb{R^3}$. How many rotations are there which take $C$ to itself? What does this question even mean? If you take any line in $\Bbb{R^3}$, and rotate the cube around it by $2\pi$ rad, you are mapping the cube to…
algebraically_speaking
  • 41
4
votes
2 answers

Can it be a valid equation if $x=-x$?

Help I don't have anybody who can help me answer my question and it is home work that is due tomorrow! I got as an end answer of $x = -x$. Is that a valid answer? If it isn't then this is the equation I started out…
Emily Chappell
  • 43
4
votes
1 answer

$f(x)$ be a polynomial with integer coefficients and $f(0) = 1989$ and $f(1) = 9891$, then no. of polynomial

Let $f(x)$ be a polynomial with integer coefficients and $f(0) = 1989$ and $f(1) = 9891$. Then prove that $f(x)$ has no integer roots. $\bf{My\; Try::}$ Let $f(x) = a_{0}x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+...........+a_{n}\;,$ where…
juantheron
  • 53,015
4
votes
3 answers

Factorising quadratics - coefficient of $x^2$ is greater than $1$

In factoring quadratics where the coefficient of $x^2$ is greater than $1$, I use the grouping method where we multiply the coefficient and constant together and then factor. My question is can someone explain the math behind…
swaggaroo
  • 41
4
votes
3 answers

Deriving two properties of clocks

This two properties are given in my module as formulas for clock related problems: $(1)$ If both the hands start moving together from the same position, both the hands will coincide after $ 65\frac5{11} $ minutes. $(2)$ Interchangeable positions…
Quixotic
  • 22,431
4
votes
2 answers

Perfect squares always one root?

I had an exam today and I was thinking about this task now, after the exam of course. $f(x)=a(x-b)^2 +c$ Now, the point was to find C so that the function only has one root. Easy enough, I played with the calculator and found this. But I hate…
Algific
  • 1,899
4
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2 answers

Convexity of Entropy function

Prove $(x+y)^{x+y}\ge x^x y^y$, subject to constraints $x>1$ and $y>1$. More generally, what other functions besides $x^x$ and $x\ln(x)$ satisfy this inequality? (I vaguely remember a paper where a well known distribution measure, which name…
Tegiri Nenashi
  • 1,030
4
votes
1 answer

high school math question, algebra, contest question.

The sum of two distinct real number is a positive integer and the sum of their squares is 2. Compute the greater of these two real numbers. I tried to set up the equations first,: $x^2 +y^2 =2\\ x+y = n,\enspace n \in \mathbb{N}$ Then I have no…
user190916
  • 125
4
votes
2 answers

If $P(x)$ is a poly. of least degree and local Max. at $x=1$ and Local Min. at $x=3,$ Then $P'(0)$ is

If $P(x)$ is a polynomial of least degree which has a local Maxima at $x=1$ and Local Minima at $x=3.$ If $P(1)=6$ and $P(3)=2$. Then $P'(0)=$ $\bf{My\; Try::}$ Given function has one Maxima and one Minima So $P(x)$ must have least degree $3$…
juantheron
  • 53,015
4
votes
2 answers

If $f(1)=1\;,f(2)=3\;,f(3)=5\;,f(4)=7\;,f(5)=9$ and $f'(2)=2,$ Then sum of all digits of $f(6)$

$(1):$ If $P(x)$ is a polynomial of Degree $4$ such that $P(-1) = P(1) = 5$ and $P(-2)=P(0)=P(2)=2\;,$Then Max. value of $P(x).$ $(2):$ If $f(x)$ is a polynomial of degree $6$ with leading Coefficient $2009.$ Suppose further that…
juantheron
  • 53,015
4
votes
2 answers

Pluggin in numbers

If $(a + c)(a − c) = 0$, which of the following must be true? $a = 0$, $c = 0$, $a = −c$, $a = c$, $a2 = c2$ The answer states: Try plugging in $a = 2$ and $c = 2$. This eliminates (A), (B), and (C). Now try $a = 2$ and $c = −2$. This eliminates…
Victor
  • 41
4
votes
6 answers

How do I solve $f(x) - f(x - 1) = x$?

Is it possible to solve the functional equation $f(x) - f(x - 1) = x$ by algebraic means?
Måns Nilsson
  • 323
4
votes
1 answer

Sempifying an algebraic expression

I have the following expressions:$$f_1=a_1^2+a_2^2+\cdots+a_n^2$$ $$f_0=(a_1x_1+a_2x_2+\cdots+a_nx_n)^2$$ $$x_1^2+x_2^2+\cdots+x_n^2=1$$ I need to show that:$$f_1-f_0\ge0$$ for all $x_i$. Assume that all not the $a_i$'s are zero.
user191392
  • 41
4
votes
1 answer

What's wrong with the calculation with polar coordinates here?

Suppose $x=r\cos t$ and $y=r\sin t$. I did the following calculation: $$ \begin{align} &x^4+y^4=(r\cos t)^4+(r\sin…
user9464
4
votes
1 answer

How do I simplify $\frac{\sqrt{1-x} + \frac{1}{\sqrt{1+x}}}{1 + \frac{1}{\sqrt{1-x}}}$?

$$\frac{\sqrt{1-x} + \frac{1}{\sqrt{1+x}}}{1 + \frac{1}{\sqrt{1-x}}}$$ I've had a go at putting everything over a common denominator in the form of: $$\frac{\frac{\sqrt{1-x}\sqrt{1+x}+1}{\sqrt{1+x}}}{\frac{\sqrt{1-x}+1}{\sqrt{1-x}}} =…
William Herschel
  • 43
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