Question

Expand each of the following, using suitable identities: 

(i) (x + 2y + 4z) 2 (ii) (2x – y + z) ² (iii) (–2x + 3y + 2z) ² 

(iv) (3a – 7b – c) ² (v) (–2x + 5y – 3z) ² (vi) [ \(\frac{1 }{ 4}\) a - \(\frac{1 }{ 2}\) b + 1]²

Answer 1

It is known that, 

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

(i) (x + 2y + 4z)² = x² (2y)² + (4z)² + 2(x) (2y) + 2 (2y) (4z) + 2(4z)(x) 

= x² + 4y² + 16z² + 4xy + 16yz + 8xz

(ii) (2x – y + z) ² = (2x)² + (-y)² + (z)2 + 2 (2x)(-y) + 2 (-y) (z) + 2 (z)(2x) 

= 4x² + y² + z² - 4xy - 2yz + 4xz

(iii) (–2x + 3y + 2z) ² = (-2x)² + (3y)² + (2z)² + 2(-2x)(3y) + 2(3y)(2z) + 2(2z)(-2x) 

= 4x² + 9y² + 4z² - 12xy + 12yz - 8xz

(iv) (–2x + 3y + 2z) 2 = (3a)2 + (-7b)² + (-c)² + 2(3a)(-7b) + 2(-7b)(-c) + 2(-c)(3a) 

= 9a² + 49b² + c² - 42ab + 14bc - 6ac

(v) (–2x + 5y – 3z) ² = (-2x)² + (5y)² + (-3z)² + 2(-2x)(5y) + 2(5y)(-3z) + 2(-3z)(-2x) 

= 4x² + 25y² + 9z² - 20xy - 30yz + 12xz

(vi) [\(\frac{1 }{ 4}\) a - \(\frac{1 }{ 2}\)2 b + 1]2 = (\(\frac{1 }{ 4}\) a)2 + (-\(\frac{1 }{ 2}\)b)² + (1)2 + 2(\(\frac{1 }{ 4}\) a)(-\(\frac{1 }{ 2}\) b) + 2 (-\(\frac{1 }{ 2}\) b)(1) + 2 (\(\frac{1 }{ 4}\) a) (1) 

= \(\frac{1 }{ 16}\) a² + \(\frac{1 }{ 4}\) b² + 1 - \(\frac{1 }{ 4}\) ab - b + \(\frac{1 }{ 2}\) a