Finding the missing value of a constant in a polynomial, using the Factor Theorem

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Adrian Jannetta

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Adrian Jannetta 7 years, 5 months ago

Created this as a copy of Finding the missing value of a constant in a polynomial, using the Factor Theorem .

Name	Status	Author	Last Modified
Finding the missing value of a constant in a polynomial, using the Factor Theorem	Ready to use	Elliott Fletcher	20/11/2019 14:39
Factor Theorem and polynomials	draft	Adrian Jannetta	16/08/2023 14:56
Finding the missing value of a constant in a polynomial, using the Factor Theorem	draft	Adrian Jannetta	16/08/2023 14:56
Marlon's copy of Finding the missing value of a constant in a polynomial, using the Factor Theorem	draft	Marlon Arcila	03/02/2018 16:11
Marlon's copy of Finding the missing value of a constant in a polynomial, using the Factor Theorem	draft	Marlon Arcila	03/02/2018 16:19
Factor Theorem	draft	Mark Hodds	24/03/2018 15:58
Dado que $({x+{d}})$ es un factor de $g(x) = ax^3+{b}x^2+{x}+m$ , Encontrar el valor de $m$ .	Ready to use	Luis Hernandez	19/06/2023 13:36
JD's copy of Finding the missing value of a constant in a polynomial, using the Factor Theorem	draft	JD Ichwan	21/05/2019 05:42
Finding the missing value of a constant in a polynomial, using the Factor Theorem	draft	Xiaodan Leng	11/07/2019 01:38

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Question statement

The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$ , then $(x-p)$ is a factor of $f(x)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

w	integer	2
a	integer	-2
b	integer	-1
d	integer	1
coef_x3	number	-2
coef_x2	number	-2
coef_x	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2,3,4)

JME function reference

Description

Random number between 2,3,4.

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

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

coef_x
coef_x2
coef_x3

Advice
"Unnamed part" - prompt
"Unnamed part" → "Unnamed gap" - Correct answer

Parts

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.

Given that $(\simplify{x+{d}})$ is a factor of $g(x) = \simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x}+k$ , find the value of $k$ .

$k =$ Unnamed gap.

Advice

Using the factor theorem, we know that if $(x-a)$ is a factor of a polynomial $f(x)$ , then $f(a)=0$ .

We are given that $(\simplify{x+{d}})$ is a factor of $g(x) = \simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x+m}$ .

By the factor theorem, this means that $g(\simplify{-{d}}) = 0$ .

Substituting $x=\simplify{-{d}}$ into $g(x)$ gives

$\begin{align} g(\simplify{-{d}}) &= \simplify[all,!collectNumbers]{{coef_x3}+{coef_x2}+{coef_x}+m}\\ &=\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}. \end{align}$

Therefore, as $g(\simplify{-{d}}) = 0$ , we have

$\begin{align} \simplify{{coef_x3}+{coef_x2}+{coef_x}+m}&=0\\ m&=\simplify{-({coef_x3}+{coef_x2}+{coef_x})}. \end{align}$

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Finding the missing value of a constant in a polynomial, using the Factor Theorem

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Adrian Jannetta 7 years, 5 months ago

Error in variable testing condition

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Generated value: `integer`

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Error

Finding the missing value of a constant in a polynomial, using the Factor Theorem

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Adrian Jannetta 7 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`