Find greatest common divisor of two polynomials over the rationals

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Let

$\begin{align}\\ f(X)&=\simplify{ X ^ 4 + {c3} * X ^ 3 + {c2} * X ^ 2 + {c1} * X + {c0}}\\\\ g(X)&=\simplify{X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}}\end{align}$

Give any introductory information the student needs.

			Name	Type	Generated Value

r1	integer	1
r2	integer	3
n	integer	5
a1	integer	3
a3	integer	1
a2	integer	2
n1	integer	5
c3	integer	9
c2	integer	35
n2	integer	15
c0	integer	66
d2	integer	6
c1	integer	73
d0	integer	21
d1	integer	16

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Consider $f(X)$ and $g(X)$ as polynomials over the rational numbers $\mathbb{Q}$ .

Find their greatest common divisor (GCD).

GCD=Unnamed gap.

You must input your answer as a monic polynomial over $\mathbb{Q}$ .

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First we divide $f(X)$ by $g(X)$ to get the remainder $r_1(X) =\simplify{ X ^ 2 + {r2 + a3} * X + {r2 * a3}}$ :

$f(X) = \simplify{X ^ 4 + {c3} * X ^ 3 + {c2} * X ^ 2 + {c1} * X + {c0} = (X + {a1}) * (X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}) + X ^ 2 + {r2 + a3} * X + {r2 * a3}}$

Next we divide $g(X)$ by $r_1(X)$ to get the remainder $r_2(X) =\simplify{ {n1} * X + {n2}}$ :

$g(X) = \simplify{X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0} = (X + {a2}) * (X ^ 2 + {r2 + a3} * X + {r2 * a3}) + {n1} * X + {n2}}$

Now divide $r_1(X)$ by $r_2(X)$ to obtain:

$r _1(X) =\simplify{ X ^ 2 + {r2 + a3} * X + {r2 * a3} = (1 / {n}) * (X + {a3}) * ({n1} * X + {n2})}$

As we have zero remainder, the last non-zero remainder is the GCD and this is $\simplify{{n1} * X + {n2}}$ .

But we have to make this into a monic polynomial and the required monic GCD $\simplify{X+{r2}}$ is given by multiplying by $\frac{1}{\var{n1}}$ so that the coefficient of the highest degree term is 1.

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Find greatest common divisor of two polynomials over the rationals

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Find greatest common divisor of two polynomials over the rationals

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`