Algebra – Harder Factorise or Factoring can be done by collecting common factors and using sums and products of factors.
$$\large x^2-(a+b)x + ab = (x-a)(x-b)$$

### Question 1

Factorise $(x^2-3x)^2-2x^2 + 6x-8$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} &= (x^2-3x)^2-2(x^2-3x)-8 \\ &= A^2-2A-8 &\color{red} \text{let } A = x^2-3x \\ &= (A + 2)(A-4) \\ &= (x^2-3x + 2)(x^2-3x-4) \\ &= (x-1)(x+2)(x-4)(x+1) \end{aligned}

### Question 2

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

\begin{aligned} \require{AMSsymbols} \displaystyle &= (x-1)(x+2) \times (x-3)(x+4) + 24 \\ &= (x^2 +x-2)(x^2 + x-12) + 24 \\ &= (A-2)(A-12) + 24 &\color{red} \text{let } A = x^2 + x \\ &= A^2-14A + 48 \\ &= (A-6)(A-8) \\ &= (x^2 + x-6)(x^2 + x-8) \\ &= (x-2)(x+3)(x^2 + x-8) \end{aligned}

### Question 3

Factorise $x^4-5x^2 +4$.

\begin{aligned} \displaystyle &= (x^2-1)(x^2-4) \\ &= (x+1)(x-1)(x+2)(x-2) \end{aligned}

### Question 4

Factorise $x^4 + x^2 + 1$.

\begin{aligned} \displaystyle &= x^4 + 2x^2 + 1-x^2 \\ &= (x^2 + 1)^2-x^2 \\ &= (x^2 + 1 + x)(x^2 + 1-x) \end{aligned}

### Question 5

Factorise $x^4-6x^2y^2 + y^4$.

\begin{aligned} \displaystyle &= x^4-2x^2y^2 + y^4-4x^2y^2 \\ &= (x^2-y^2)^2-(2xy)^2 \\ &= (x^2-y^2 + 2xy)(x^2-y^2-2xy) \end{aligned}

### Question 6

Factorise $x^3 + 2x^2y^2-x-2y$.

\begin{aligned} \displaystyle &= 2(x^2-1)y + x(x^2-1) \\ &= (x^2-1)(2y + x) \\ &= (x+1)(x-1)(2y+x) \end{aligned}

### Question 7

Factorise $x^3 + x^2z + xz^2-y^3-y^2z-yz^2$.

\begin{aligned} \displaystyle &= (x-y)z^2 + (x^2-y^2)z + x^3-y^3 \\ &= (x-y)z^2 + (x-y)(x + y)z + (x-y)(x^2 + xy + y^2) \\ &= (x-y)\big[z^2 + (x + y)z + x^2 + xy + y^2\big] \\ &= (x-y)(z^2 + xz + yz + x^2 + xy + y^2) \\ &= (x-y)(x^2 + y^2 + z^2 + xy + yz + zx) \end{aligned}

### Question 8

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

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} &= (A-8x)(A-7x)-6x^2 &\color{red} \text{let } A = x^2 + 12 \\ &= A^2-15Ax + 56x^2-6x^2 \\ &= A^2-15Ax + 50x^2 \\ &= (A-10x)(A-5x) \\ &= (x^2-5x + 12)(x^2-10x + 12) \end{aligned}

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