Polynomials: Expand and Simplify 2c

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Created this as a copy of Polynomials: Expand and Simplify 1a.

Name	Status	Author	Last Modified
Polynomials: Expand and Simplify 1a	Ready to use	Ben McGovern	14/08/2023 15:48
Polynomials: Expand and Simplify 1b	Ready to use	Ben McGovern	14/08/2023 15:52
Polynomials: Expand and Simplify 1c	Ready to use	Ben McGovern	14/08/2023 15:53
Polynomials: Expand and Simplify 1d	Ready to use	Ben McGovern	14/08/2023 16:15
Polynomials: Expand and Simplify 1e	Ready to use	Ben McGovern	14/08/2023 16:17
Polynomials: Expand and Simplify 2c	Ready to use	Ben McGovern	14/08/2023 16:21
Polynomials: Expand and Simplify 2b	Ready to use	Ben McGovern	14/08/2023 16:21
Polynomials: Expand and Simplify 2d	Ready to use	Ben McGovern	14/08/2023 16:21
Cubics: Finding a Quadratic Factor 1	Ready to use	Ben McGovern	14/09/2022 15:45
Cubics: Finding Linear Factors	Ready to use	Ben McGovern	14/08/2023 15:39
Polynomials: Expand and Simplify 2a	Ready to use	Ben McGovern	14/08/2023 16:19

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Question statement

Given three functions,

$f(x) = \simplify{{a1}x^2+{b1}x+{c1}}, \quad g(x)=\simplify{{b2}x+{c2}}, \quad h(x)=\simplify{{b3}x+{c3}}$

calculate the product

$\simplify{f(x)g(x)h(x)}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	integer	1
b1	integer	5
c1	integer	7
b2	integer	1
c2	integer	-2
b3	integer	4
c3	integer	2

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Advice

To find the product of the three functions $f(x)$ , $g(x)$ , and $h(x)$ , we can first find the product of two of the functions, and then multiply the resulting polynomial by the third function. For the functions

$f(x) = \simplify{{a1}x^2+{b1}x+{c1}}, \quad g(x)=\simplify{{b2}x+{c2}}, \quad h(x)=\simplify{{b3}x+{c3}},$

since $g(x)$ and $h(x)$ are linear, we will find their product first:

$\begin{split} g(x)h(x) &,=(\simplify{{b2}x+{c2}})(\simplify{{b3}x+{c3}}) \\ &\,= \simplify[!cancelTerms]{{b2*b3}x^2+{b2*c3}x+{b3*c2}x+{c2*c3}} \\ &\,=\simplify{{b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3}} \end{split}$

We can now find multiply this result by $f(x)$ to find the product of all three functions:

$\begin{split} f(x)g(x)h(x) &\,= (\simplify{{a1}x^2+{b1}x+{c1}})(\simplify{{b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3}}) \\ &\,= \simplify[all]{{a1}x^2({b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3})+{b1}x({b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3})+{c1}({b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3})}. \end{split}$

Expanding each bracket and collecting similar terms:

$\begin{split} f(x)g(x)h(x) &\,= \simplify[all]{{a1}x^2({b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3})+{b1}x({b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3})+{c1}({b2*b3}x^2+{b2*c3+b3*c2}x+{c2*c3})} \\ &\,=\simplify[all,!collectNumbers, !cancelTerms]{{a1*b2*b3}x^4+{a1*b2*c3+a1*b3*c2}x^3+{a1*c2*c3}x^2+{b1*b2*b3}x^3+{b1*b2*c3+b1*b3*c2}x^2+{b1*c2*c3}x+{c1*b2*b3}x^2+{c1*b2*c3+c1*b3*c2}x+{c1*c2*c3}} \\ &\,= \simplify{{a1*b2*b3}x^4+{a1*b2*c3+a1*b3*c2}x^3+{a1*c2*c3}x^2+{b1*b2*b3}x^3+{b1*b2*c3+b1*b3*c2}x^2+{b1*c2*c3}x+{c1*b2*b3}x^2+{c1*b2*c3+c1*b3*c2}x+{c1*c2*c3}}. \end{split}$

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This question is used in the following exams:

Expanding and Simplifying Polynomials by Ben McGovern in MASH Bath: Moodle quizzes and TS.
All the good stuff by Chris Templet in Chris's workspace.
Exam test by Paula Severi in Paula's workspace.

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Polynomials: Expand and Simplify 2c

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Taxonomy: mathcentre

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Taxonomy: Context

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Ben McGovern 2 years, 6 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 4 years, 8 months ago

Error in variable testing condition

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Generated value: integer

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Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

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Generated value: `integer`