JD's copy of Use the factor theorem to identify factors of a polynomial

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JD Ichwan 5 years, 10 months ago

Created this as a copy of Use the factor theorem to identify factors of a polynomial.

Name	Status	Author	Last Modified
Use the factor theorem to identify factors of a polynomial	Ready to use	Elliott Fletcher	20/11/2019 14:44
Use the factor theorem to identify factors of a polynomial	draft	Adrian Jannetta	16/08/2023 14:56
Marlon's copy of Use the factor theorem to identify factors of a polynomial	draft	Marlon Arcila	03/02/2018 16:13
Amy's copy of Use the factor theorem to identify factors of a polynomial	draft	Amy Fracarossi	28/06/2018 01:10
El teorema del factor establece que si $f (x)$ es un polinomio y $f (p) = 0$ , entonces $(xp)$ es un factor de $f (x)$ .	Ready to use	Luis Hernandez	19/06/2023 13:22
JD's copy of Use the factor theorem to identify factors of a polynomial	draft	JD Ichwan	21/05/2019 05:42
Use the factor theorem to identify factors of a polynomial	draft	Xiaodan Leng	11/07/2019 01:44

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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

a	integer	-1
b	integer	-1
c	integer	3
d	integer	2
coef1_x3	number	1
coef1_x2	number	1
coef1_x	integer	-5
const	integer	3
coef2_x3	number	-8
coef2_x2	number	4
coef2_x	integer	10
coef3_x3	number	-27
coef3_x2	number	9
coef3_x	integer	15

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xxxxxxxxxx
 
random(-2..3 except 0 except c)

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Random number between -2 and 3, not including 0 for creating polynomial.

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Generated value: `integer`

-1

← Depends on:

→ Used by:

coef1_x
coef1_x2
coef1_x3
coef2_x
coef2_x2
coef3_x
coef3_x2
const
d

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Use the factor theorem to find which two of the following are factors of the polynomial

$f(x) = \simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.$

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$(\simplify{x+{a}})$

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$(\simplify{x+{d}})$

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$(\simplify{x+{c}})$

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Advice

To find the factors of the polynomial $f(x) = \simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}$ , we use the factor theorem.

If $f(x)$ is a polynomial and $f(p) = 0$ , then $(x-p)$ is a factor of $f(x)$ .

If $(\simplify{(x+{a})})$ is a factor of $f(x)$ then by the factor theorem, $f(\simplify{-{a}}) = 0$ .

We see that

$\begin{align} f(\simplify{-{a}}) &= \simplify[all,!collectNumbers]{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}\\ &= \simplify{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}. \end{align}$

Therefore, $(\simplify{(x+{a})})$ is a factor of $f(x)$ .

Similarly for $(\simplify{(x+{d})})$ ,

$\begin{align} f(\simplify{-{d}}) &= \simplify[all,!collectNumbers]{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\ &= \simplify{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\ &\neq 0. \end{align}$

Therefore, $(\simplify{(x+{d})})$ is not a factor of $f(x)$ .

Finally, for $(\simplify{(x+{c})})$ ,

$\begin{align} f(\simplify{-{c}}) &= \simplify[all,!collectNumbers]{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}\\ &= \simplify{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}. \end{align}$

Therefore, $(\simplify{(x+{c})})$ is also a factor of $f(x)$ .

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JD's copy of Use the factor theorem to identify factors of a polynomial

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Taxonomy: mathcentre

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JD Ichwan 5 years, 10 months ago

Error in variable testing condition

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Generated value: `integer`

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JD's copy of Use the factor theorem to identify factors of a polynomial

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

JD Ichwan 5 years, 10 months ago

Error in variable testing condition

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Generated value: integer

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Choose several from a list

Advanced settings

Generated value: `integer`