Factorising Harder Quadratics

Harder Factorise

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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