Harder Factorise


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

Question 1

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

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

Question 2

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

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

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

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

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