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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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a problem on a special root of $x^{11}-1=0$

I have came across the following problem if $\alpha$ be a special root of the equation $x^{11}-1=0$ , then prove that $$(\alpha+1)(\alpha^2+1)......(\alpha^{10}+1)=1$$ totally stuck on it. how to solve this.please help me somebody.
user59908
  • 231
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Solve $4+\frac{1}{x}-\frac{1}{x^2}$ using quadratic formula

I am to solve for x using the quadratic formula: $$4+\frac{1}{x}-\frac{1}{x^2}=0$$ The solution provided in the answers section is: $\dfrac{-1\pm\sqrt{17}}{8}$ whereas I arrived at something entirely different:…
Doug Fir
  • 2,266
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Existence of maps on $\mathbb{N} \cup \{0\}$ satisfying $\phi(ab)=\phi(a)+\phi(b)$

How many maps $\phi : \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} $ are there with the property that $\phi(ab)=\phi(a)+\phi(b)$, for all $a,b \in \mathbb{N} \cup \{0\} $? My Attempt is $$\phi(0)+\phi(m)=\phi(0) \implies \phi(m)=0\quad \text{…
user408906
6
votes
5 answers

How to find the last digit of $37^{100}$

I am currently working on some algebra and I am studying the modulus chapter of my book. One question was finding the last digit of: $$37^{100}$$ They give a hint about how the solution should be about calculating a remainder... Is there any easy…
Lukas Arvidsson
  • 2,376
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if $m<\frac{1}{2}$, what is $-\frac{1}{m}$ equal to?

I start out with: $$m<\frac{1}{2}$$ Taking the reciprocal of both side flips inequality sign: $$\frac{1}{m}>2$$ Multiply both side by -1 flips the sign yet again: $$-\frac{1}{m}<-2$$. But the result $-\frac{1}{m}<-2$ is not valid. If $m=-100$, then…
most venerable sir
  • 1,689
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Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$?

Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$ ? $f(x)=x^4+x^3+x^2+x+1$ $f(x^5)=x^{20}+x^{15}+x^{10}+x^5+1$
HOLYBIBLETHE
  • 2,770
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4 answers

Equation $x^{\frac{n+1}{n}}=x+1$

Let $n$ be a positive integer. What is the positive value of $x$ such that $x^{\frac{n+1}{n}}=x+1$? This equation has a unique solution because the function $x^{\frac{n+1}{n}}-x$ is increasing. However, I'm not sure if we can get a closed form for…
user11550
  • 568
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Alternative for calculating the nth of quadratic sequence

Given the quadratic sequence $$f(n)=1, 7, 19, 37, \cdots$$ To calculate the $f(n)$ for $n\ge1$. $$f(n)=an^2+bn+c$$ We start with the general quadratic function, then sub in for $n:=1,2$ and $3$ $$f(1)=a+b+c$$ $$f(2)=4a+2b+c$$ $$f(3)=9a+3b+c$$ Now…
user546240
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1 answer

How can I re-arrange this equation?

I haven't used my algebra skills much for years and they seem to have atrophied significantly! I'm having real trouble working out how to re-arrange a formula I've come across to get $x$ by itself on the left hand side. It looks like…
user8170
6
votes
3 answers

What is the formula for this curve?

Three years of calculus in college have served me nothing, apparently, since I can't for the life of me remember even the basics. I'm working on a small software project where I have a table with say 20 cells, and I want the cells' opacity to go…
Snowman
  • 2,664
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Solving a system of quadratic equations.

Solve for real $(a, b, c)$ satisfying $$ab + bc + ca = 1$$ $$a^2 − 2b^2 = 1$$ $$2b^2 − 3c^2 = 1$$ I try isolating $a$, but it leads to a very complicated expression in $a$.
John Chang
  • 758
6
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5 answers

If a seller sold half the amount and $\frac12$ a unit to each customer, and ended up selling the entire stock. How many units were sold?

The seller was asked how many cheese pieces he had sold. He replied: "Today there were $4$ buyers, each buyer bought half of the remaining cheese pieces and half of one cheese." As a result, all cheeses was sold." How many cheeses has been sold? All…
Rami Chasygov
  • 248
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5 answers

How would I solve $x^2-4x=y^2-4y$ without knowing the answer beforehand?

The equation is $x^2-4x=y^2-4y$ in the case where $x\ne y$. The answer is $x+y=4$. I can start from $x+y=4$ and create the equation very easily, and I can substitute $x+4=y$ into the equation and show both sides are equal easily. I just don't get…
Hockeyfan19
  • 643
6
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2 answers

If $a,b,c,d$ be the roots of the biquadratic $x^4-x^3+2x^2+x+1=0$ then show that $(a^3+1)(b^3+1)(c^3+1) (d^3+1)=16$

If $a,b,c,d$ be the roots of the biquadratic $x^4-x^3+2x^2+x+1=0$ then show that $(a^3+1)(b^3+1)(c^3+1) (d^3+1)=16$ I have tried to solve the equation first and find the values of the roots but it becomes very long process. Is there any easy…
pakupa
  • 61
6
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7 answers

Solving $\frac{1}{(x-1)} - \frac{1}{(x-2)} = \frac{1}{(x-3)} - \frac{1}{(x-4)}$. Why is my solution wrong?

I'm following all hitherto me known rules for solving equations, but the result is wrong. Please explain why my approach is not correct. We want to solve: $$\frac{1}{(x-1)} - \frac{1}{(x-2)} = \frac{1}{(x-3)} - \frac{1}{(x-4)}\tag1$$ Moving the…
saner
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