Cubics: Finding Linear Factors

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Created this as a copy of Polynomials: Expand and Simplify 2b.

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

Factorise the following cubic expression into a product of linear factors, where one of the factors is $\simplify{{b1}x+{c1}}$ :

$\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}}$

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			Name	Type	Generated Value

b1	integer	4
c1	integer	2
b2	integer	2
c2	integer	2
b3	integer	4
c3	integer	5

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If the cubic expression

$\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}}$

can be factorised into a product of linear factors, and we know that one factor is $\simplify{{b1}x+{c1}}$ , it is easiest to first find the remaining quadratic factor. We can then factorise this quadratic into two linear factors.

We can start by taking a generalised quadratic in the form $ax^2+bx+c$ as the quadratic factor:

$\begin{split}\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}} &\,= (\simplify{{b1}x+{c1}})(ax^2+bx+c) \\ &\,= \simplify{{b1}x(a*x^2+b*x+c) + {c1}(a*x^2+b*x+c)}. \end{split}$

If we now expand the brackets and collect similar terms, we can compare coefficients to find the values of $a$ , $b$ , and $c$ :

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

By comparing the coefficients of each side of the equation we can create a set of simultaneous equations to find $a$ , $b$ , and $c$ :

$\begin{split} & \simplify{{b1*b2*b3}} &\,= \simplify{{b1}a} \\ &\simplify{{c1*b2*b3+c2*b1*b3+c3*b1*b2}} &\,= \simplify{{b1}b+{c1}a} \\ &\simplify{{b1*c2*c3+b2*c1*c3+b3*c1*c2}} &\,=\simplify{{b1}c+{c1}b} \\ &\simplify{{c1*c2*c3}} &\,=\simplify{{c1}c}. \end{split}$

Hence,

$a=\var{b2*b3},\quad b=\var{b1},\quad c=\var{c1}.$

Therefore,

$\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}} = (\simplify{{b1}x+{c1}})(\simplify{{b2*b3}x^2+{c2*b3+b2*c3}x+{c2*c3}}).$

Finally, we need to factorise the quadratic $\simplify{{b2*b3}x^2+{c2*b3+b2*c3}x+{c2*c3}}$ :

$\simplify{{b2*b3}x^2+{c2*b3+b2*c3}x+{c2*c3}}=(\simplify{{b2}x+{c2}})(\simplify{{b3}x+{c3}}).$

Thus, the cubic expression as a product of three linear factors is

$\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}}=(\simplify{{b1}x+{c1}})(\simplify{{b2}x+{c2}})(\simplify{{b3}x+{c3}}).$

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This question is used in the following exams:

Algebra basics by Ruth Hand in MASH Bath: Moodle quizzes and TS.
Cubic Equations by Ben McGovern in MASH Bath: Moodle quizzes and TS.
Polynomial functions by Jinhua Mathias in Maths 3-2020.
All the good stuff by Chris Templet in Chris's workspace.
Unit 1 Test by Stephen Abreu in Stephen 's workspace.

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Cubics: Finding Linear Factors

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Taxonomy: mathcentre

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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

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Generated value: integer

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Generated value: `integer`