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Question
Factorize the following polynomial.
(x2 – 6x)2 – 8 (x2 – 6x + 8) – 64
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Solution
(x2 – 6x)2 – 8 (x2 – 6x + 8) – 64
= (x2 – 6x)2 – 8(x2 – 6x) – 64 – 64
= (x2 – 6x)2 – 8(x2 – 6x) – 128
Let x2 – 6x = z.
∴ (x2 – 6x)2 – 8(x2 – 6x) – 128
= z2 – 8z – 128
= z2 – 16 z + 8z – 128
= z(z – 16) + 8 (z – 16)
= (z – 16) (z +8)
= (x2 – 6x – 16) (x2 – 6x + 8) ...(Replace z = x2 – 6x)
= (x2 – 8x + 2x – 16) (x2 – 4x – 2x + 8)
= [x (x – 8) + 2 (x – 8)] [x (x – 4) – 2 (x – 4)]
= (x – 8) (x + 2) (x – 4) (x – 2)
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