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michaelcai
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michaelcai
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617
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102
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19
115 Questions
19 Answers
+1
842
1
+617
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 $x+y+z$.
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michaelcai
20.02.2018
+3
1675
3
+617
Last logarithm problems
Solve and find the domain of the equations:
and
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michaelcai
12.02.2018
+3
721
1
+617
Logarithm problem
Solve and find the domain of the equation:
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michaelcai
12.02.2018
+3
812
2
+617
Help
Solve and find the domain of the equation:
heureka
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michaelcai
12.02.2018
+3
756
2
+617
Help
Solve and find the domain of the equation:
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michaelcai
12.02.2018
0
5426
2
+617
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.$$If $t$ is an integer and $f(t)>5$, what is the smallest possible value of $f(t)$?
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michaelcai
29.01.2018
+1
4611
7
+617
Problem: If $f$ is a polynomial of degree $4$ such that $$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
29.01.2018
0
1060
1
+617
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 $(x-1)(x-2)(x-3)$
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michaelcai
26.01.2018
0
1118
1
+617
Evaluate the infinite geometric series:
Evaluate the infinite geometric series:
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michaelcai
25.01.2018
0
4825
1
+617
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 $x^3$.
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michaelcai
23.01.2018
0
4737
1
+617
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 $f(0)$.
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michaelcai
23.01.2018
0
4502
1
+617
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.
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michaelcai
23.01.2018
+1
4007
2
+617
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 $(x-1)(x-2)(x-3)$.
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michaelcai
23.01.2018
0
2942
4
+617
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.
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michaelcai
20.12.2017
+2
2829
6
+617
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
14.12.2017
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