**Binomial expansion formula for 1 plus x whole power n :**

**Here we are going to see the formula for the b**inomial expansion formula for 1 plus x whole power n.

(1 + x)^{n}

(1 - x)^{n}

(1 + x)^{-n}

(1 - x)^{-n}

**Example 1 :**

Write the first four terms in the expansion of (1 + 4x)^{-5} where |x| < 1/4

**Solution :**

| 4x | = 4 | x | < 4 (1/4) = 1 4x | < 1

(1 + 4x)^{− 5} can be expanded by Binomial theorem.

x = 4x , n = 5

= 1 - 5 (4x) + (5 (5+1)/2!) (4x)^{2}

- (5 (5+1) (5+2)/3!) (4x)^{3 }+ .........

= 1 - 20x + 15 (16x^{2}) - 35 (64x^{3}) + .........

= 1 - 20x + 240x^{2} - 2240x^{3} + .........

**Example 2 :**

Write the first four terms in the expansion of (1 - x^{2})^{-4} where |x| < 1

**Solution :**

(1 - x^{2})^{-4} can be expanded by binomial theorem since

|x^{2}| < 1

= 1 + 4 (x^{2}) + (4 (4+1) /2!)(x^{2})^{2}+ (4 (4+1) (4+2)/3!)(x^{2})^{3}+ ...........

= 1 + 4 (x^{2}) + (4x5/2!)x^{4}+ (4x5x6/3!)x^{6}+ ...........

= 1 + 4 x^{2} + 10 x^{4}+ 20 x^{6}+ ...........

**Example 3 :**

Write the first four terms in the expansion of (1 - x^{2})^{-4} where |x| < 1

**Solution :**

(1 - x^{2})^{-4} can be expanded by binomial theorem since

|x^{2}| < 1

= 1 + 4 (x^{2}) + (4 (4+1) /2!)(x^{2})^{2}+ (4 (4+1) (4+2)/3!)(x^{2})^{3}+ ...........

= 1 + 4 (x^{2}) + (4x5/2!)x^{4}+ (4x5x6/3!)x^{6}+ ...........

= 1 + 4 x^{2} + 10 x^{4}+ 20 x^{6}+ ...........

**Example 4 :**

Find the expansion of 1/(2 + x)^{4} where |x| < 2 upto the fourth term.

**Solution :**

1/(2 + x)^{4 }= (2 + x)^{-4 }= 2^{-4 }(1 + x/2)^{-4 }can be expanded by binomial theorem since

|x/2| < 1

**Example 5 :**

Find the value of ∛126 correct to two decimal places.

**Solution :**

∛126 = (126)^{1/3}

= (125 + 1)^{1/3 }

= [125 (1 + (1/125))]^{1/3}

= 125^{1/3}(1 + (1/125))^{1/3 } 1/125 < 1

= 5 [1 + (1/3)(1/125) + ..........]

= 5 [1 + (1/3)(0.008)]

= 5 [1 + 0.002666]

= 5.01 (correct to 2 decimal places)

After having gone through the stuff given above, we hope that the students would have understood, "Binomial expansion formula for 1 plus x whole power n".

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