Solve: x^3-3x^2+27 is congruent to 0 (mod 225)

Question
Polynomial factorization
asked 2021-01-30
Solve:
\(\displaystyle{x}^{{3}}-{3}{x}^{{2}}+{27}\) is congruent to 0 (mod 225)

Answers (1)

2021-01-31
Step 1
Consider the following problem \(\displaystyle{x}^{{3}}−{3}{x}^{{2}}+{27}\equiv{0}{\left(\text{mod}{225}\right)}\)
Prime factorization of 225 is \(\displaystyle{3}^{{2}}\times{5}^{{2}}\)
Now, \(\displaystyle{225}=\frac{{{3}^{{2}}\times{5}^{{2}}}}{{{x}^{{3}}-{3}{x}^{{2}}+{27}}}\) implies that the polynomial is reducible over \(\displaystyle{F}_{{3}}{\quad\text{and}\quad}{F}_{{5}}\)
So, there is a solution under 3 with x=0 and a solution under modulo 5 with x=1
Since the modulus is composite , so we solve the equation with \(\displaystyle\text{mod}{\left({3}^{{2}}\times{5}^{{2}}\right)}\)
Therefore, we have
x=0(mod 3)
x=1(mod 5)
Step 2
Now, using Chinese remainder theorem for
x=0(mod 3)
x=1(mod 5)
we get x=6
0

Relevant Questions

asked 2021-02-21
Solve. The factorization of the polynomial is \((3x+2)(2x^{2}+1)\).
Given Information:
The provided polynomial is \(6x^{3}+4x^{2}+3x+2\).
asked 2021-01-06
Need to calculate:The factorization of \(x^{3}+8x^{2}-3x-24\).
asked 2021-03-09
Need to calculate:The factorization of \(x^{3}+4x^{2}+3x+12\).
asked 2020-12-24
Need to calculate:The factorization of \(x^{3}+3x^2+2x+6\)
asked 2021-01-22
Does the equation \(\displaystyle{x}^{{2}}\equiv{x}\cdot{x}\equiv{2}{x}\cdot{4}{x}\text{mod}{7}\) show that factorization of polynomials mod 7 is not unique? Why or why not?
asked 2020-12-28
Find the roots of the given quadratic equation by factorization method
\(\displaystyle{x}^{{2}}-{3}{x}-{10}={0}\)
asked 2020-11-27
Write P into factored form
\(\displaystyle{P}{\left({x}\right)}={x}^{{3}}+{3}{x}^{{2}}+{4}{x}+{12}\)
asked 2021-01-07
Need to calculate:The factorization of \(3x^{3}+2x^{2}+3x+2\).
asked 2021-02-24
Use the factorization theorem to determine whether \(\displaystyle{x}−\frac{{1}}{{2}}\) is a factor
of \(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{4}}−{x}^{{3}}+{2}{x}−{1}\).
asked 2021-02-16
Use the factor theorem to determine if the given binomial is a factor of f(x).
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}+{8}{x}^{{3}}+{11}{x}^{{2}}-{11}{x}+{3},{x}+{3}\)
...