Factorising Cubic Expressions with Rotating Three Variables such as \( a^2+a^2c+ac^2-b^3-b^2c-bc^2 \)

Factorising Cubic Expressions with Rotating Three Variables such as \( a^2+a^2c+ac^2-b^3-b^2c-bc^2 \)

For these sorts of factorisations involving rotation of variables, it would be a good idea to rearrange regarding only a specific variable. Example 1 Factorise \( a^2+a^2c+ac^2-b^3-b^2c-bc^2 \). \( \begin{align} &= (ac^2-b^c)+(a^2c-b^2c)+(a^3-b^3) \\ &= (a-b)c^2+(a^2-b^2)c+(a^3-b^3) \\ &= (a-b)c^2+(a-b)(a+b)c+(a-b)(a^2+ab+b^2) \\ &= (a-b)\left[ c^2+(a+b)c+(a^2+ab+b^2) \right] \\ &= (a-b)(a^2+b^2+c^2+ab+bc+ca) \end{align} \) Example 2 Factorise \( x^3+3ax^2+(3a^2-b^2)x+a^3-ab^2 \). \( […]

Factorising Quartic Expressions with two Quadratic Factors and a Remainder, such as \( (x^2+3x-2)(x^2+3x+4)-27 \) \( (x^2-8x+12)(x^2-7x+12)-6x^2 \)

Factorising Quartic Expressions with two Quadratic Factors and a Remainder, such as \( (x^2+3x-2)(x^2+3x+4)-27 \) \( (x^2-8x+12)(x^2-7x+12)-6x^2 \)

Example 1 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 2 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 \\ &= […]

Factorising Quartics with Four Factors and a Remainder such as \( (x-1)(x-3)(x+2)(x+4)+24 \)

Factorising Quartics with Four Factors and a Remainder such as \( (x-1)(x-3)(x+2)(x+4)+24 \)

Example 1 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} &= […]