Factorisation Made Easy: Overcoming Harder Expressions

Factorising Cubic Expressions with Rotating Three Variables

Factorising cubic expressions is a crucial skill in algebra, and it becomes even more intriguing when dealing with rotating three variables. In this guide, we’ll explore the intricate process of factorising cubic expressions while optimizing for the search term “factorising cubic expressions.”

Importance of Factorising Cubic Expressions

1. Problem Solving: Factorising cubic expressions is essential for solving equations and simplifying complex mathematical problems.

2. Identifying Roots: Factoring helps identify the roots or solutions of cubic equations, providing valuable insights into their behaviour.

Factorising Cubic Expressions with Rotating Three Variables

1. Begin with Common Factors: Look for common factors among the expressions’ terms, such as constants or variables.

2. Grouping: Group terms with common factors and factor them separately.

3. Factor by Grouping: Apply techniques like grouping, difference of squares, or perfect cubes to factor the expression further.

4. Use Trial and Error: If necessary, use trial and error to find additional factors or roots.

5. The Role of Rotating Variables: Consider how each variable’s presence affects the factorisation process when rotating three variables. Rotate the variables to simplify the expression and reveal factorisation patterns.

Example Problems

1. Basic Cubic Expression: Factorising a simple cubic expression to find its roots.

2. Rotating Three Variables: Tackling a more complex cubic expression with variables in constant rotation.

Factorising cubic expressions, especially with rotating three variables, is a challenging yet rewarding aspect of algebra. This guide provides you with essential tools and techniques to master this skill. Whether solving equations or simplifying complex expressions, a strong grasp of factorisation will be a valuable mathematical asset.

It would be a good idea to rearrange regarding only a specific variable for these sorts of factorisations involving rotation of variables.

Example 1

Factorise \( a^3+a^2c+ac^2-b^3-b^2c-bc^2 \).

\( \begin{align} &= (ac^2-bc^2)+(a^2c-b^2c)+(a^3-b^3) \\ &= (a-b)c^2+(a^2-b^2)c+(a^3-b^3) \\ &= (a-b)c^2+(a-b)(a+b)c+(a-b)(a^2+ab+b^2) \\ &= (a-b)\left[ c^2+(a+b)c+(a^2+ab+b^2) \right] \\ &= (a-b)(a^2+b^2+c^2+ab+bc+ca) \end{align} \)

Example 2

Factorise \( x^3+3ax^2+(3a^2-b^2)x+a^3-ab^2 \).

\( \begin{align} &=(x^3+a^3)+(3ax^2+3a^2x)-(b^2x+ab^2) \\ &= (x+a)(x^2-ax+a^2)+3ax(x+a)-b^2(x+a) \\ &= (x+a)(x^2-ax+a^2+3ax-b^2) \\ &= (x+a)(x^2+2ax+a^2-b^2) \\ &= (x+a)\left[ (x+a)^2-b^2 \right] \\ &= (x+a)(x+a+b)(x+a-b) \end{align} \)

Example 3

Factorise \( a^2(b-c)+b^2(c-a)+c^2(a-b) \).

\( \begin{align} &=(b-c)a^2 + b^2c-ab^2 +ac^2-bc^2 \\ &= (b-c)a^2+(-ab^2+ac^2) +(b^2c-bc^2) \\ &= (b-c)a^2-(b^2-c^2)a +(b-c)bc \\ &= (b-c)a^2-(b-c)(b+c)a +(b-c)bc \\ &= (b-c) \left[ a^2-(b+c)a+bc \right] \\ &= (b-c)(a-b)(a-c) \end{align} \)

Example 4

Factorise \( a^2+b^2+c^2-2ab-2bc+2ac \).

\( \begin{align} &= a^2+(-2ab+2ac)+b^2+c^2-2bc \\ &= a^2-2(b-c)a+(b-c)^2 \\ &= \left[ a-(b-c)^2 \right] \\ &= (a-b+c)^2 \end{align} \)


Factorising Quartic Expressions with two Quadratic Factors and a Remainder

Example 5

Factorise \( (x^2+3x-2)(x^2+3x+4)-27 \).

\( \require{AMSsymbols} \begin{align} &= \left[ (\bbox[yellow]{x^2+3x})-2 \right] \left[ (\bbox[yellow]{x^2+3x})+4 \right]-27 \\ &= (\bbox[yellow]{x^2+3x})^2 + 2(\bbox[yellow]{x^2+3x})-8-27 \\ &= (\bbox[yellow]{x^2+3x})^2 + 2(\bbox[yellow]{x^2+3x})-35 \\ &= (\bbox[yellow]{x^2+3x}+7)(\bbox[yellow]{x^2+3x}-5) \end{align} \)

Example 6

Factorise \( (x^2-8x+12)(x^2-7x+12)-6x^2 \).

\( \require{AMSsymbols} \begin{align} &= \left[ (x^2+12)-8x \right] \left[ (x^2+12)-7x \right]-6x^2 \\ &= (x^2+12)^2 -15x(x^2+12) + 56x^2-6x^2 \\ &= (x^2+12)^2\bbox[aqua]{-15x(x^2+12)} + 50x^2 \leadsto 50x^2 = \bbox[pink]{-10x} \times \bbox[yellow]{-5x} \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow]{x^2+12} &\bbox[pink]{-10x} &\bbox[pink]{-10x(x^2+12)} \\ &\bbox[pink]{x^2+12} &\bbox[yellow]{-5x} &\bbox[yellow]{-5x(x^2+12)} \\ \hline &&&\bbox[aqua]{-15x(x^2+12)} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow]{x^2+12} \bbox[pink,3px]{-10x})(\bbox[pink]{x^2+12}\bbox[yellow,3px]{-5x}) \\ &= (x^2-10x+12)(x^2-5x+12) \end{align} \)


Factorising Quartics with Four Factors and a Remainder

Example 7

Factorise \( (x-1)(x-3)(x+2)(x+4)+24 \).

\( \require{AMSsymbols} \begin{align} &= (x-1)(x+2) \times (x-3)(x+4) + 24 \\ &= (x^2+x-2) \times (x^2+x-12) +24 \\ &= \left[(x^2+x)-2\right] \times \left[(x^2+x)-12\right] +24 \\ &= (x^2+x)^2-14(x^2+x)+24+24 \\ &= (x^2+x)^2\bbox[aqua]{-14(x^2+x)}+48 \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow]{x^2+x} &\bbox[pink,3px]{-6} &\bbox[pink]{-6(x^2+x)} \\ &\bbox[pink]{x^2+x} &\bbox[yellow,3px]{-8} &\bbox[yellow]{-8(x^2+x)} \\ \hline &&&\bbox[aqua]{-14(x^2+x)} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow]{x^2+x}\bbox[pink,3px]{-6})(\bbox[pink]{x^2+x}\bbox[yellow,3px]{-8}) \\ &= (x-2)(x+3)(x^2+x-8) \end{align} \)

Example 8

Factorise \( (x+1)(x+2)(x-3)(x-4)+6 \).

\( \require{AMSsymbols} \begin{align} &= (x+1)(x-3) \times (x+2)(x-4)+6 \\ &= (x^2-2x-3) \times (x^2-2x-8)+6 \\ &= \left[(x^2-2x)-3\right] \times \left[(x^2-2x)-8\right]+6 \\ &= (x^2-2x)^2-11(x^2-2x)+24+6 \\ &= (x^2-2x)^2\bbox[aqua]{-11(x^2-2x)}+30 \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow]{x^2-2x} &\bbox[pink,3px]{-5} &\bbox[pink]{-5(x^2-2x)} \\ &\bbox[pink]{x^2-2x} &\bbox[yellow,3px]{-6} &\bbox[yellow]{-6(x^2-2x)} \\ \hline &&&\bbox[aqua]{-11(x^2-2x)} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= \left[\bbox[yellow]{(x^2-2x)}\bbox[pink,5px]{-5}\right] \left[\bbox[pink]{(x^2-2x)}\bbox[yellow,5px]{-6}\right] \\ &= (x^2-2x-5)(x^2-2x-6) \end{align} \)


Factorising Quadratics with Six Terms of \( x \) and \(y \)

Sometimes, students may encounter complex quadratics factorise involving \( x^2, y^2, xy, x \) and \( y \). Consider factorising by only either \( x \) or \( y \). The following examples take through factorising \( y \) first, then \( x\).

Example 9

Factorise \( x^2 + 2xy + 5x + y^2+ 5y + 6 \).

\( \require{AMSsymbols} \begin{align} &= x^2 + 2xy + 5x + (y^2 + 5y+6) \\ &= x^2+\bbox[aqua,3px]{2xy + 5x}+(y+2)(y+3) \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow,5px]{x} &\bbox[pink]{(y+2)} &\bbox[pink]{x(y+2)=xy+2x} \\ &\bbox[pink,5px]{x} &\bbox[yellow]{(y+3)} &\bbox[yellow]{x(y+3)=xy+3x} \\ \hline &&&\bbox[aqua,3px]{2xy+5x} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow,5px]x+\bbox[pink]{(y+2)})(\bbox[pink,5px]x+\bbox[yellow]{(y+3)}) \\ &= (x+y+2)(x+y+3) \end{align} \)

Example 10

Factorise \( x^2-3xy-3x+2y^2+4y+2 \).

\( \require{AMSsymbols} \begin{align} &= x^2-3xy-3x+2(y^2+2y+1) \\ &= x^2\bbox[aqua,3px]{-3xy-3x}+2(y+1)(y+1) \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow,5px]{x} &\bbox[pink]{-(y+1)} &\bbox[pink]{-x(y+1)=-xy-x} \\ &\bbox[pink,5px]{x} &\bbox[yellow]{-2(y+1)} &\bbox[yellow]{-2x(y+1)=-2xy-2x} \\ \hline &&&\bbox[aqua,3px]{-3xy-3x} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow,5px]x\bbox[pink]{-(y+1)})(\bbox[pink,5px]x\bbox[yellow]{-2(y+1)}) \\ &= (x-y-1)(x-2y-2) \end{align} \)

Example 11

Factorise \( x^2+3xy+5x+2y^2+7y+6 \).

\( \require{AMSsymbols} \begin{align} &= x^2+3xy+5x+(2y^2+7y+6) \\ &= x^2+\bbox[aqua,3px]{3xy+5x}+(2y+3)(y+2) \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow,5px]{x} &\bbox[pink]{(2y+3)} &\bbox[pink]{x(2y+3)=2xy+3x} \\ &\bbox[pink,5px]{x} &\bbox[yellow]{(y+2)} &\bbox[yellow]{x(y+2)=xy+2x} \\ \hline &&&\bbox[aqua,3px]{3xy+5x} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow,5px]x+\bbox[pink]{(2y+3)})(\bbox[pink,5px]x+\bbox[yellow]{(y+2)}) \\ &= (x+2y+3)(x+y+2) \end{align} \)

Example 12

Factorise \( x^2-2xy+x-3y^2+5y-2 \).

\( \require{AMSsymbols} \begin{align} &= x^2-2xy+x-(3y^2-5y+2) \\ &= x^2\bbox[aqua,3px]{-2xy+x}-(3y-2)(y-1) \end{align} \)

\( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow,5px]{x} &\bbox[pink]{-(3y-2)} &\bbox[pink]{-x(3y-2)=-3xy+2x} \\ &\bbox[pink,5px]{x} &\bbox[yellow]{(y-1)} &\bbox[yellow]{x(y-1)=xy-x} \\ \hline &&&\bbox[aqua,3px]{-2xy+x} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow,5px]x\bbox[pink]{-(3y-2)})(\bbox[pink,5px]x+\bbox[yellow]{(y-1)}) \\ &= (x-3y+2)(x+y-1) \end{align} \)

Example 13

Factorise \( 2x^2-7xy+11x+3y^2-13y+12 \).

\( \require{AMSsymbols} \begin{align} &= x^2-7xy+11x+(3y^2-13y+12) \\ &= 2x^2\bbox[aqua,3px]{-7xy+11x}+(3y-4)(y-3) \end{align} \)

\( \require{AMSsymbols} \begin{array} {rcr} &\bbox[yellow,5px]{x} &\bbox[pink]{-(3y-4)} &\bbox[pink]{-2x(3y-4)=-6xy+8x} \\ &\bbox[pink,5px]{2x} &\bbox[yellow]{-(y-3)} &\bbox[yellow]{-x(y-3)=-xy+3x} \\ \hline &&&\bbox[aqua,3px]{-7xy+11x} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow,5px]x\bbox[pink]{-(3y-4)})(\bbox[pink,5px]{2x}\bbox[yellow]{-(y-3)}) \\ &= (x-3y+4)(2x-y+3) \end{align} \)

Example 14

Factorise \( 12x^2+20xy-24x-8y^2+22y-15 \).

\( \require{AMSsymbols} \begin{align} &= 12x^2+20xy-24x-(8y^2-22y+15) \\ &= 12x^2+\bbox[aqua,3px]{20xy-24x}-(4y-5)(2y-3) \end{align} \)

\( \require{AMSsymbols} \begin{array} {rcr} &\bbox[yellow,5px]{2x} &\bbox[pink]{(4y-5)} &\bbox[pink]{6x(4y-5)=24xy-30x} \\ &\bbox[pink,5px]{6x} &\bbox[yellow]{-(2y-3)} &\bbox[yellow]{-2x(2y-3)=-4xy+6x} \\ \hline &&&\bbox[aqua,3px]{20xy-24x} \end{array} \)

\( \require{AMSsymbols} \begin{align} &= (\bbox[yellow,5px]{2x}+\bbox[pink]{(4y-5)})(\bbox[pink,5px]{6x}\bbox[yellow]{-(2y-3)}) \\ &= (2x+4y-5)(6x-2y+3) \end{align} \)

Conclusion

Factorisation is a powerful tool that can turn the most intimidating expressions into manageable equations. Armed with easy factorization techniques, you can confidently overcome harder expressions. Embrace the challenge, practice diligently, and watch as factorization transforms from a daunting task into a valuable skill in your mathematical arsenal. Mathematics just got a little easier!

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