Add polynomials in Z_n

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

Compute the following sums of polynomials.

Give any introductory information the student needs.

			Name	Type	Generated Value

ans1

integer

			Name	Type	Generated Value

b0	integer	4
b4	integer	2
b2	integer	2
c0	integer	3
c1	integer	1
c2	integer	1
c3	integer	3
c4	integer	0
c5	integer	2
d0	integer	2
d1	integer	1
d2	integer	3
d3	integer	3
d4	integer	2
d5	integer	2
d6	integer	2

			Name	Type	Generated Value

f0	integer	6
f1	integer	2
f2	integer	5
f3	integer	4
f4	integer	6
f5	integer	1
g	integer	1
g0	integer	6
g1	integer	2
g2	integer	3
g3	integer	3
g4	integer	5
g5	integer	3
h	integer	6
k0	integer	0
k1	integer	0
k2	integer	2
k3	integer	1
k4	integer	1
k5	integer	5
k6	integer	0

			Name	Type	Generated Value

a1	integer	1
a2	integer	2
a3	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
mod(-a1-a2-a3,3)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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Part c) Gap-fill
1. Gap Mathematical expression
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Find the following sum of polynomials over $\mathbb{Z}_3$ .

All coefficients of non-zero terms in your answer should be either $1$ or $2$ .

$(X - \var{ a1}) + (X - \var{ a2}) + (X - \var{ a3}) =$ Unnamed gap

Advice

a)

Adding the polynomials we obtain $3X-\var{a1+a2+a3}$ . But $3 \equiv 0 \pmod 3$ and $-\var{a1+a2+a3} \equiv \var{ans1} \pmod 3$ .

Hence the sum is $\var{ans1}$ .

b)

Summing up, we obtain:

$\begin{align} (\simplify{X ^ 6 + {b4} * X ^ 4 + {b2} * X ^ 2 + {b0}})&+\simplify{ {g}*(X ^ 6 + {c5} * X ^ 5 + {c4} * X ^ 4 + {c3} * X ^ 3 + {c2} * X ^ 2 + {c1} * X + {c0})} \\ &= \simplify{{1 + g} * X ^ 6 + {g * c5} * X ^ 5 + {b4 + g * c4} * X ^ 4 + {g * c3} * X ^ 3 + {b2 + g * c2} * X ^ 2 + {g * c1} * X + {b0 + g * c0}} \\ &\equiv \simplify{{d6} * X ^ 6 + {d5} * X ^ 5 + {d4} * X ^ 4 + {d3} * X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}} \pmod 5 \end{align}$

c)

Once again, on summing up we obtain:

$\begin{align} \simplify{X ^ 6 + {f5} * X ^ 5 + {f4} * X ^ 4 + {f3} * X ^ 3 + {f2} * X ^ 2 + {f1} * X + {f0} } &+ \simplify{ {h}*(X ^ 6 + {g5} * X ^ 5 + {g4} * X ^ 4 + {g3} * X ^ 3 + {g2} * X ^ 2 + {g1} * X + {g0})} \\ &= \simplify{ {1 + h} * X ^ 6 + {f5 + h * g5} * X ^ 5 + {f4 + h * g4} * X ^ 4 + {f3 + h * g3} * X ^ 3 + {f2 + h * g2} * X ^ 2 + {f1 + h * g1} * X + {f0 + h * g0}} \\ &\equiv \simplify{{k6} * X ^ 6 + {k5} * X ^ 5 + {k4} * X ^ 4 + {k3} * X ^ 3 + {k2} * X ^ 2 + {k1} * X + {k0}} \pmod 7 \end{align}$

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Taxonomy: mathcentre

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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Add polynomials in Z_n

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`