# Polynomials

## Exercise 2.5 Part 6

Question: 5 – Factorise:

(i) 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz – 16xz

Answer: Given, 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz – 16xz

= (2x)^2 + (3y)^2 + ( - 4z)^2

 + 2(2x)xx(3y) + 2(3y)xx(-4z) + 2(2x)xx( - 4z)

If a = 2x, b = 3y and c = - 4z
Then, using the identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac, we get;

4x^2 + 9y^2 + 16z^2 + 12xy - 24yz – 16xz

= (2x + 3y – 4z)(2x + 3y – 4z)

= (2x + 3y – 4z)^2

(ii) 2x^2 + y^2 + 8z^2 - 2sqrt2xy + 4sqrt2yz – 8xz

Answer: Given, 2x^2 + y^2 + 8z^2 - 2sqrt2xy + 4sqrt2yz – 8xz
This expression can be written as follows:
(-sqrt2x)^2 + y^2 + (2sqrt2z)^2

+2( - sqrt2x)xx(z) + 2(y)xx(2sqrt2z) + 2(sqrt2x)xx(2sqrt2z)

If, a = -sqrt2x, b = y and c = 2sqrt2z
Then, using the identity (a + b + c)^2

= a^2 + b^2 + c^2 + 2ab + 2bc + 2ac

We get;
2x^2 + y^2 + 8z^2 - 2sqrt2xy + 4sqrt2yz – 8xz

= (- sqrt2x + y + 2sqrt2z)( - sqrt2x + y + 2sqrt2z)

= (- sqrt2x + y + 2sqrt2z)^2