Divide Polynomials: Cubic by linear

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Ben Brawn

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Ben Brawn 8 years, 10 months ago

Created this as a copy of Divide Polynomials.

Name	Status	Author	Last Modified
Divide Polynomials	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Divide Polynomials: Cubic by linear	draft	Ben Brawn	11/09/2017 04:05
cormac's copy of Divide Polynomials	draft	cormac breen	30/10/2018 15:53
Luis's copy of Divide Polynomials	draft	Luis Hernandez	30/11/2018 01:09
Luis's copy of Divide Polynomials	draft	Luis Hernandez	30/11/2018 14:55
Dividir un polinomio cúbico $p(x)=ax^3+bx^2+cx+d$ por $q(x)=mx+n$ , para hallar el cociente y el resto.	Ready to use	Luis Hernandez	19/06/2023 13:34
Divide Polynomials	draft	Xiaodan Leng	10/07/2019 23:21

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Question statement

Divide $\displaystyle{\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}$ by $\simplify[std]{{r}x+{s}}$ so that:

$\frac{\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\simplify[std]{{r}x+{s}}}=q(x)+\frac{r}{\simplify[std]{{r}x+{s}}}$

where $q(x)$ is the quotient polynomial and $r$ is the remainder ( $r$ is a constant).

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

Give any introductory information the student needs.

			Name	Type	Generated Value

be	integer	-3
s2	integer	-1
s1	integer	1
m	integer	1
al	integer	-9
n	integer	-4
s	integer	6
r	integer	1
t	integer	-3

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Description

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Generated value: `integer`

-3

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Parts

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

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

Input all numbers as integers and not as decimals.

$r=\;\;$ Gap 1.

Advice

There is a procedure called polynomial long division, if you are comfortable with long division of numbers then it isn't too different. You can see an example of the procedure here. However, if you don't recall long division we can rewrite the numerator so that it includes a multiple of the denominator. We do this term by term starting with the leading term (that is, $\simplify{{r * m}x ^ 3}$ ).

$\begin{eqnarray*} \simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}&=&\simplify[std]{(x+{s})x^2+{n}x^2+{n*s+t}x+{t*n+be}}\\&=&\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+{t}x+{t*n+be}}\\ &=&\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+(x+{s})*{t}+{t*n+be-s*t}}\\ &=&\simplify[std]{(x+{s})(x^2+{n}x+{t})+{t*n+be-s*t}} \end{eqnarray*}$

Hence

$\frac{\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\simplify[std]{{r}x+{s}}}=\simplify[std]{x^2+{n}x+{t}+{t*n+be-s*t}/({r}x+{s})}$

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Ben Brawn 8 years, 10 months ago

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Generated value: `integer`

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Divide Polynomials: Cubic by linear

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Ben Brawn 8 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`