michaelcai

115 Questions
19 Answers

+1

846

1

+618

Determine a constant $k$ such that the polynomial$$ P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$is divisible by $x+y+z$.

Determine a constant $k$ such that the polynomial$$ P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$is divisible by $x+y+z$.

michaelcai Feb 20, 2018

+3

1677

3

+618

Last logarithm problems

Solve and find the domain of the equations:

and
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michaelcai Feb 12, 2018

+3

725

1

+618

Logarithm problem

Solve and find the domain of the equation:

michaelcai Feb 12, 2018

+3

817

2

+618

Help

Solve and find the domain of the equation:

michaelcai Feb 12, 2018

+3

759

2

+618

Help

Solve and find the domain of the equation:

michaelcai Feb 12, 2018

0

5427

2

+618

Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that$$f(p) = f(q) = f(r) = f(s

Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that$$f(p) = f(q) = f(r) = f(s) = 5.$$If $t$ is an integer and $f(t)>5$, what is the smallest possible value of $f(t)$?

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michaelcai Jan 29, 2018

+1

4612

7

+618

Problem: If $f$ is a polynomial of degree $4$ such that $$f(0) = f(1) = f(2) = f(3) = 1$$ and $$f(4) = 0,$$ then determine $f(5)$.

If $f$ is a polynomial of degree $4$ such that $$f(0) = f(1) = f(2) = f(3) = 1$$ and $$f(4) = 0,$$ then determine $f(5)$.

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michaelcai Jan 29, 2018

0

1064

1

+618

Suppose $f(x)$ is a polynomial of degree $4$ or greater such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the remainder when $f(x)$ is divid

Suppose $f(x)$ is a polynomial of degree $4$ or greater such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the remainder when $f(x)$ is divided by $(x-1)(x-2)(x-3)$

michaelcai Jan 26, 2018

0

1122

1

+618

Evaluate the infinite geometric series:

Evaluate the infinite geometric series:

michaelcai Jan 25, 2018

0

4825

1

+618

Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)-1$ is divisible by $(x-1)^3$. $\bullet$ $f(x

Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold:
$f(x)-1$ is divisible by $(x-1)^3$.
$f(x)$ is divisible by $x^3$.

michaelcai Jan 23, 2018

0

4739

1

+618

Suppose the polynomial $f(x)$ is of degree $3$ and satisfies $f(3)=2$, $f(4)=4$, $f(5)=-3$, and $f(6)=8$. Determine the value of $f(0)$.

Suppose the polynomial $f(x)$ is of degree $3$ and satisfies $f(3)=2$, $f(4)=4$, $f(5)=-3$, and $f(6)=8$. Determine the value of $f(0)$.

michaelcai Jan 23, 2018

0

4504

1

+618

Find all integers $n$ for which $\frac{n^2+n+1}{n-1}$ is an integer.

Find all integers $n$ for which $\frac{n^2+n+1}{n-1}$ is an integer.

michaelcai Jan 23, 2018

+1

4012

2

+618

Suppose $f(x)$ is a polynomial of degree $4$ or greater such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the remainder when $f(x)$ is divide

Suppose $f(x)$ is a polynomial of degree $4$ or greater such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the remainder when $f(x)$ is divided by $(x-1)(x-2)(x-3)$.

michaelcai Jan 23, 2018

0

2944

4

+618

Let $f(x)=ax^2+bx+a$, where $a$ and $b$ are constants and $a\ne 0$. If one of the roots of the equation $f(x)=0$ is $x=4$, what is the other

Let $f(x)=ax^2+bx+a$, where $a$ and $b$ are constants and $a\ne 0$. If one of the roots of the equation $f(x)=0$ is $x=4$, what is the other root? Explain your answer.

michaelcai Dec 20, 2017

+2

2832

6

+618

Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.

Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.

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michaelcai Dec 14, 2017