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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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Bounding $(x+y)^n$

Let $n$ be a natural number. Is it possible to write $$(x+y)^n \leq C(x^n + y^n)$$ for some constant $C$?? It is obvious for $n=2$ (using Young's inequality) but not obvious to me for other $n$. Let $x$ and $y$ be positive reals.
LapLace
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Solve for $x$ in $6x^2-25x+12+\frac{25}{x}+\frac{6}{x^2}=0$

I simplified to get $6x^4-25x^3+12x^2+25x+6=0$ Or $6 (x^2+1)^2+25x(1-x^2)=0$ Then I substituted $z=x^2+1$, to get $6z^2+25\sqrt{(z-1)}(2-z)=0$ I can't find a way to proceed further.
Adienl
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Which is bigger: $\sqrt{1001} - \sqrt{1000}$, or $\frac{1}{10}$?

Which is bigger: $\sqrt{1001} - \sqrt{1000}$, or $\frac{1}{10}$? I can calculate the answer using a calculator, however I suspect to do so may be missing the point of the question. The problem appears in a book immediately after a section called…
mikoyan
  • 1,135
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Infinite progression - done right?

To solve: $ \lim\limits_{n\to \infty}(x+x^3+x^5+...+x^{2n-1})=-\frac{2}{3}$. So we see that it's a geometric progression (constant quotient) and it's going on infinitely. So we can apply the formula: $\sum_{n=0}^\infty a_1 q^n=\frac{a_1}{1-q}$ and…
Bringiton
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Prove that $\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$

Let $a,b,c$ are positive real numbers such that, $a^2+b^2+c^2=3$. Prove that $$\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$$
Drona
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Express $\;f(x)=\frac{x − 1}{x + 1}\;$ as the sum of an even and an odd function.

In homework there is such problem: Express $\;f(x)=\dfrac{x − 1}{x + 1}\;$ as the sum of an even and an odd function. (Simplify as much as possible.) This function is not even and neither odd. Also if we take it as division of 2 functions,…
Tsiskreli
  • 63
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Precalculus Project decision

OK so I have to do a research paper/presentation on an experiment/project that relates to my precalculus class. Only problem is that I was given no topics to choose from and I couldn't find any real good ones online. Can anybody give me some good…
Ronnie.j
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Are there real numbers such that ...

$x^2 + xy =3$ and $x - y^2 = 2$? I graphed it and there are no intersections so obviously there is no real number solutions, but is there a "mathier" (read algebraic) way to prove this?
jd123
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On factorizing and solving the polynomial: $x^{101} – 3x^{100} + 2x^{99} + x^{98} – 3x^{97} + 2x^{96} + \cdots + x^2 – 3x + 2 = 0$

The actual problem is to find the product of all the real roots of this equation,I am stuck with his factorization: $$x^{101} – 3x^{100} + 2x^{99} + x^{98} – 3x^{97} + 2x^{96} + \cdots + x^2 – 3x + 2 = 0$$ By just guessing I noticed that $(x^2 – 3x…
Quixotic
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Is it possible to have a quadratic equation having only one complex root? If so, what would a picture of it look like?

I'm also wondering the same questions about a quadratic function with two real roots, and a quadratic function with two pure imaginary roots. Is it possible? And if it is possible, what would it look like when graphed? I'm trying to fully understand…
Jordan
  • 123
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Square Root of $x^2$

Does the $\sqrt{x^2}$ equal $x$ or $|x|$? In other words is it an agreement that we take by default the positive square root of $x$ or we have to explicitly define it with absolute value as there could be two roots to $x^2$ ($+x$ or $-x$)?
Infinity
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Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$

I am stuck with the following problem: Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$Let $\,m$ be the number of distinct complex non-real roots and $\,n$ be the number of distinct real roots of the above equation. Then $\,m-n\,$…
learner
  • 6,726
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Express $a^n+b^n$ in terms of $x$ and $y$ where $a+b=x$ and $ab=y$

If $a+b=x$ and $ab=y$, express $a^n+b^n$ in terms of $x$ and $y$. The following may help you find the pattern. \begin{align*} a + b &= x\\ a^2 + b^2 &= x^2 - 2 y\\ a^3 + b^3 &= x^3 - 3 x y\\ a^4 + b^4 &= x^4 - 4 x^2 y + 2 y^2\\ a^5 + b^5 &= x^5 - 5…
kiss my armpit
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Intuition behind problem in (classical) algebra

To give some background, the question is to show that if $a=b+c$ then $$a^4+b^4+c^4 = 2a^2b^2+2b^2c^2+2c^2a^2$$ Which, for completeness, I was able to do by squaring twice $$(a-b-c)^2=0$$ gives $$a^2+b^2+c^2= 2(bc-ac-ab)$$ which on squaring gives…
kuch nahi
  • 6,789
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condition for a cubic polynomial to have a real root

Let $a,b \in R$ and assume that $x=1$ is a root of the polynomial $p(x)= x^4+ax^3+bx^2+ax+1$. Find the ranges of values of $a$ for which $p$ has a complex root which is not real. Here first I factored out $x-1$ which left me with a cubic…
tattwamasi amrutam
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