Question

Factorize the following polynomials. 

i. (x² – x)² – 8(x² – x) + 12

ii. (x – 5)² – (5x – 25) – 24

iii. (x² – 6x)² – 8(x² – 6x + 8) – 64

iv. (x² – 2x + 3) (x² – 2x + 5) – 35

v. (y + 2) (y – 3) (y + 8) (y + 3) + 56

vi. (y² + 5y) (y² + 5y – 2) – 24

vii. (x – 3) (x – 4)² (x – 5) – 6

Answer 1

i. (x² – x)² – 8(x² – x) + 12 

= m² – 8m + 12 …[Putting x² – x = m] 

= m² – 6m – 2m + 12 

= m(m – 6) – 2(m – 6) 

= (m – 6)(m – 2) 

= (x² – x - 6) (x² – x - 2) …[Replacing m = x² - x] 

= (x² – 3x + 2x – 6) (x² – 2x + x – 2) 

= [x(x – 3) + 2(x – 3)] [x(x – 2) + 1 (x - 2)] 

= (x – 3) (x + 2) (x – 2) (x + 1)

ii. (x – 5)² – (5x – 25) – 24 

= (x – 5)² – (5x – 25) – 24 

= (x – 5)² – 5(x – 5) – 24 

= m² – 5m – 24 … [Putting x – 5 = m] 

= m² – 8m + 3m – 24 

= m(m – 8) + 3(m – 8) 

= (m – 8) (m + 3)

= (x – 5 – 8) (x – 5 + 3) … [Replacing m = x – 5] 

= (x – 13) (x – 2)

iii. (x² – 6x)² – 8(x² – 6x + 8) – 64 

= m² – 8(m + 8) - 64 …[Putting x² – 6x = m] 

= m² – 8m – 64 – 64 

= m² – 8m – 128 

= m² – 16m + 8m - 128 

= m(m – 16) + 8(m – 16) 

= (m – 16)(m + 8) 

= (x² – 6x – 16) (x² – 6x + 8) … [Replacing m = 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)

iv. (x² – 2x + 3) (x² – 2x + 5) – 35 

= (m + 3) (m + 5) – 35 … [Putting x² – 2x = m] 

= m (m + 5) + 3(m + 5) – 35 

= m² + 5m + 3m + 15 – 35 

= m² + 8m – 20 

= m² + 10m – 2m – 20 

= m(m + 10) – 2(m + 10) 

= (m + 10) (m – 2) 

= (x² – 2x + 10) (x² – 2x – 2) … [Replacing m = x² – 2x]

v. (y + 2) (y – 3) (y + 8) (y + 3) + 56 

= (y + 2)(y + 3)(y – 3)(y + 8) + 56 

= (y² + 3y + 2y + 6) (y² + 8y – 3y – 24) + 56 

= (y² + 5y + 6) (y² + 5y – 24) + 56

= (m + 6) (m – 24) + 56 … [Putting y² + 5y = m] 

= m (m – 24) + 6 (m – 24) + 56 

= m² – 24m + 6m – 144 + 56 

= m² – 18m – 88 

= m² – 22m + 4m – 88 

= m(m – 22) + 4(m – 22) 

= (m – 22) (m + 4) 

= (y² + 5y – 22)(y² + 5y + 4) … [Replacing m = y² + 5y] 

= (y² + 5y – 22) (y² + 4y + y + 4) 

= (y² + 5y – 22) [y(y + 4) + 1(y + 4)] 

= (y² + 5y – 22) (y + 4) (y + 1)

vi. (y² + 5y) (y² + 5y – 2) – 24 

= (m)(m – 2) – 24 … [Putting y² + 5y = m] 

= m² – 2m – 24 

= m² – 6m + 4m – 24 

= m(m – 6) + 4(m – 6) 

= (m – 6) (m + 4) 

= (y² + 5y – 6) (y² + 5y + 4) … [Replacing m = y² + 5y]

= (y² + 6y – y – 6) (y² + 4y + y + 4) 

= [y(y + 6) – 1(y + 6)] [y(y + 4) + 1(y + 4)] 

= (y + 6) (y – 1) (y + 4) (y + 1)

vii. (x – 3) (x – 4)² (x – 5) – 6 

= (x – 3) (x – 5) (x – 4)² – 6 

= (x² – 5x – 3x + 15) (x² – 8x + 16) – 6 

= (x² – 8x + 15) (x² – 8x + 16) – 6 

= (m + 15) (m+ 16) – 6 … [Putting x² – 8x = m] 

= m (m + 16) + 15 (m + 16) – 6 

= m² + 16m + 15m + 240 – 6 

= m² + 31m + 234 

= m² + 18m + 13m + 234 

= m(m + 18) + 13(m + 18) 

= (m + 18) (m + 13) 

= (x² – 8x + 18) (x² – 8x + 13) … [Replacing m = x² – 8x]