Factorise polynomial into irreducible factors

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Name	Status	Author	Last Modified
Factorise polynomial into irreducible factors	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Marlon's copy of Factorise polynomial into irreducible factors	draft	Marlon Arcila	03/02/2018 16:39
Luis's copy of Factorise polynomial into irreducible factors	draft	Luis Hernandez	30/11/2018 18:56

Question statement

Let

$\begin{align}\\ f(X)&=\simplify{ X ^ 4 + {f3} * X ^ 3 + {f2} * X ^ 2 + {f1} * X + {f0}}\\\\ g(X)&=\simplify{X^5+{d4}*X^4+{d3}*X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}}\end{align}$

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			Name	Type	Generated Value

f0	integer	2
f1	integer	3
f2	integer	3
f3	integer	4
h0	integer	1
h1	integer	4
al	integer	4
b0	integer	3
b1	integer	1
ga	integer	3
d4	integer	4
d2	integer	1
d3	integer	2
d0	integer	1
d1	integer	0
be	integer	0
g1	integer	0
g0	integer	2
a1	integer	2
a2	integer	4
c2	integer	1
c1	integer	2
c0	integer	1
a	integer	2
c	integer	4
b	integer	4
n	integer	5

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xxxxxxxxxx
 
mod(h0*g0,n)

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Generated value: `integer`

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Consider $f(X)$ and $g(X)$ as polynomials over $\mathbb{Z}_{\var{n}}$ .

Find their greatest common divisor (GCD) and enter it as a monic polynomial.

Note that the coefficients all should be in $\mathbb{Z}_{\var{n}}=\{0,\;1,\;2,\;3,\;4\}$ .

GCD=Unnamed gap.

Click on Show steps to obtain information on monic polynomials over $\mathbb{Z}_{\var{n}}$ .

Advice

1. We have on using the Euclidean Algorithm that the GCD of $f(X)$ and $g(X)$ is $\simplify{X^2+{g1}*X+{g0}}$ over $\mathbb{Z}_{\var{n}}$ .

Note that this GCD is a monic polynomial.

2. Hence $\simplify{X^2+{g1}*X+{g0}}$ is a factor of $f(X)$ and is irreducible over $\mathbb{Z}_{\var{n}}$ as you can check.

On dividing $f(X)$ by $\simplify{X^2+{g1}*X+{g0}}$ we obtain:

$f(X) = \simplify{(X ^ 2 + {g1} * X + {g0}) * (X ^ 2 + {h1} * X + {h0})}$ over $\mathbb{Z}_{\var{n}}$ .

You can check that $\simplify{X ^ 2 + {h1} * X + {h0}}$ is also irreducible over $\mathbb{Z}_{\var{n}}$ .

Hence the above factorization expresses $f(X)$ as the product of irreducible polynomials over $\mathbb{Z}_{\var{n}}$ .

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Factorise polynomial into irreducible factors

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Factorise polynomial into irreducible factors

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Gap-fill

Generated value: `integer`