**Example 1:**

Factorize the given polynomials to the limit they can be factorized.

**a)** x$^{4}$ – 81

**b)** (x – 3)$^{2}$ – (y – 2)$^{2}$

**c)** Z$^{8}$ – 1

**d)** x$^{2}$ – 81z$^{2}$

**Solution:**

**a)** x$^{4}$ – 81 = (x$^{2})^{2}$ – 9$^{2}$

= (x$^{2}$ – 9) (x$^{2}$ + 9)

= (x$^{2}$ – 3$^{2}$) (x$^{2}$ + 9)

= (x + 3) (x – 3) (x$^{2}$ + 9)

**b)** (x – 3)$^{2}$ – (y – 2)$^{2}$ = (x – 3 + y – 2) {x – 3 – (y – 2)}

= (x + y – 5) (x – y – 1)

**c)** z$^{8}$ – 1 = (z$^{4})^{2}$ – 1$^{2}$

= (z$^{4}$ – 1) (z$^{4}$ + 1)

= {(z$^{2})^{2}$ – 1$^{2}$} (z$^{4}$ + 1)

= (z$^{2}$ – 1) (z$^{2}$ + 1) (z$^{4}$ + 1)

= (z + 1) (z – 1) (z$^{2}$ + 1) (z$^{4}$ + 1)

**d)** x$^{2}$ – 81z$^{2}$ = x$^{2}$ – (9y)$^{2}$

= (x – 9y) (x + 9y)

**Example 2:**

Find the value of the given numbers

**a)** 97 x 103

**b)** 33 x 27

**c)** 49 x 51

**Solution:**

**a)** 97 x 103

97 = 100 – 3

103 = 100 + 3

So, 97 x 103 = (100 – 3) (100 + 3) = (100)$^{2}$ – 3$^{2}$ = 10000 – 9 = 9991

**b)** 32 x 28

32 = 30 + 2

28 = 30 – 2

32 x 28 = (30 + 2) (30 – 2) = (30)$^{2}$ – 2$^{2}$ = 900 – 4 = 896

**c)** 49 x 51

49 = 50 – 1

51 = 50 + 1

49 x 51 = (50 – 1) (50 + 1) = (50)^2 – 1 = 2500 – 1 = 2499