Dividing polynomials 3

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

Divide $\displaystyle{f(x)=\simplify[std]{ x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1}}}$ by $g(x)=\simplify[std]{x^2+{e1}}$ so that:

$\frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}$

where $q(x)$ is the quotient polynomial and $r(x)$ and the degree of $r(x)$ is less than the degree of $g(x)$ .

The coefficients of $q(x)$ are integers, do not input as decimals.

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	integer	-4
s	integer	1
b1	integer	-1
c1	integer	-4
e1	integer	1
d1	integer	-4

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Description

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

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Generated value: `integer`

-4

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Gap-fill

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

$q(x)=\;\;$ Gap 0.

$r(x)=\;\;$ Gap 1.

Input all numbers as integers and not as decimals in both questions.

Advice

You can use the method of long division. Here we show an alternative method.

The question tells us we can write:

$\frac{f(x)}{g(x)}=\simplify[std]{(x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1})/(x^2+{e1})=q(x)+r(x)/(x^2+{e1})}\;\;\;\dots(1)$
Since the degree of

$f(x)$ is

$4$ and the degree of

$g(x)$ is

$2$ , the degree of

$q(x)$ is

$2$ and we can write

$q(x)=Ax^2+Bx+C$ for constants

$A,\;B,\;C$ .

The degree of $r(x)$ is less than $2$ hence $r(x)=Dx+E$ for constants $D,\;E$ .

Multiplying both sides of (1) by $\simplify[std]{x^2+{e1}}$ gives:

$\simplify[std]{x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1}=(Ax^2+Bx+C)(x^2+{e1})+Dx}+E$
We can then compare coefficients to determine the constants

$A,\;B,\;C,\;D,\;E$ .

Terms in $x^4$

$A=1$

Terms in $x^3$

$B=\var{a1}$

Terms in $x^2$

$\simplify[std]{{e1}A+C = {b1}} \Rightarrow C = \var{b1-e1}$

Terms in $x$

$\simplify[std]{{e1}B+D = {c1}} \Rightarrow D = \var{c1-a1*e1}$

Constant terms

$\simplify[std]{{e1}C}+E= \var{d1} \Rightarrow E = \var{d1-(b1-e1)*e1}$

Therefore:

$\frac{f(x)}{g(x)}=\simplify[std]{x ^ 2 + {a1} *x + {b1 -e1}+({c1-a1*e1}*x+{d1-e1*(b1-e1)})/(x^2+{e1})}$

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This question is used in the following exams:

mathcentre: Polynomial division by Bill Foster in Bill's workspace.
Maths Support: Polynomial division by Christian Lawson-Perfect in Maths Support Wiki.
Dividing polynomials 1 by Bill Foster in Bill's workspace.
Bradford: Dividing polynomials 1 by Bill Foster in Bill's workspace.
Algebra: 1 by Aneela Arshad in Aneela's workspace.
Harry's copy of Maths Support: Polynomial division by Harry Flynn in Harry's workspace.

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Taxonomy: mathcentre

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Contributors

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

Terms in $x^4$

Terms in $x^3$

Terms in $x^2$

Terms in $x$

Constant terms

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Dividing polynomials 3

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Terms in x4x^4

Terms in x3x^3

Terms in x2x^2

Terms in xx

Constant terms

Generated value: `integer`

Terms in $x^4$

Terms in $x^3$

Terms in $x^2$

Terms in $x$