Question

Using binomial theorem, expand each of the following: (1 + 2x – 3x²)⁴

Answer 1

To find: Expansion of (1 + 2x – 3x²)⁴ 

Formula used: (i) ⁿC_r = \(\frac{n!}{(n-r)!(r)!}\) 

(ii) (a+b)ⁿ = ⁿC_0aⁿ + ⁿC_1a ^n-1b + ⁿC_2a ^n-2b² + …… +ⁿC_n-1ab^n-1 + ⁿC_nbⁿ 

We have, (1 + 2x – 3x²)⁴ 

Let (1+2x) = a and (-3x²) = b … (i) 

Now the equation becomes (a + b)⁴ 

(Substituting value of b from eqn. i)

⇒ 1 + 8x + 24x² + 32x³ + 16x⁴ 

We have (1+2x)⁴ = 1 + 8x + 24x² + 32x³ + 16x⁴ … (iii) 

For (a+b)³ , we have formula a³+b³+3a²b+3ab²

For, (1+2x)³ , substituting a = 1 and b = 2x in the above formula 

⇒ 1³ + (2x)³ + 3(1)² (2x) +3(1) (2x)² 

⇒ 1 + 8x³ + 6x + 12x² 

⇒ 8x³ + 12x² + 6x + 1 … (iv) 

For (a+b)² , we have formula a²+2ab+b²

For, (1+2x)² , substituting a = 1 and b = 2x in the above formula 

⇒ (1)² + 2(1)(2x) + (2x)² 

⇒ 1 + 4x + 4x² 

⇒ 4x² + 4x + 1 … (v) 

Putting the value obtained from eqn. (iii),(iv) and (v) in eqn. (ii)

⇒ 1 + 8x + 24x² + 32x³ + 16x⁴ - 96x⁵ - 144x⁴ - 72x³ - 12x² + 216x⁶ + 216x⁵ + 54x⁴ - 108x⁶ - 216x⁷ + 81x⁸ 

On rearranging

81x⁸ - 216x⁷ + 108x⁶ + 120x⁵ - 74x⁴ - 40x³ + 12x² +8x+ 1