Simplify each of the following expressions :
(i) (x+y+z)²+ (x+y/2+z/3)² - (x/2+y/3+z/4)²
(ii) (x+y-2z)²-x²– y² - 3z²+4xy
(iii) (x² – x + 1)² - (x² + x + 1)²
Answers
(i) Given : (x + y + z)²+ (x + y/2 + z/3)² - (x/2 + y/3 + z/4)²
By using an identity (a + b + c)² = a² + b² + c² + 2ab + 2 bc + 2 ca)
= [(x² + y² + z² + 2xy + 2y z + 2 xz)] + [(x² + (y/2)² + (z/3)² + 2x × y/2 + 2y/2 × z/3 + 2 x × z/3)] - [(x/2)² + (y/3)² + (z/4)² + 2x/2 × y/3 + 2y/3 × z/4 + 2 x/2 × z/4) ]
= [(x² + y² + z² + 2xy + 2y z + 2 xz)] + [(x² + y²/4) + z²/9 + xy + yz/3 + 2xz/3)] - [x²/4 + y²/9 + z²/16 + xy/3 + yz/6 + xz/4) ]
= x² + y² + z² + 2xy + 2yz + 2xz + x² + y²/4 + z²/9 + xy + yz/3 + 2xz/3 - x²/4 - y²/9 - z²/16 - xy/3 - yz/6 - xz/4
= (2x² - x²/4) + (y² + y²/4 - y²/9) + (z² + z²/9 - z²/16) + ( 2xy + xy - xy/3) + ( 2yz + yz/3 - yz/6) + (2xz + 2xz/3 - xz/4)
= (8x² - x²)/4 + (36y² + 9y² - 4y²)/36 + (144z² + 16z²- 9z²)/144) + (9xy - xy)/3 + (13yz)/6 + (29xz )/12
= 7x²/4 + 41y²/36 + 151z²/144 + 8xy/3 + 13yz/6 + 29xz/12
(ii) Given : (x + y - 2z)² - x² - y² - 3z² + 4xy
By using an identity (a + b + c)² = a² + b² + c² + 2ab + 2 bc + 2 ca)
= (x² + y² + (- 2z)² )+ 2xy + 2 × y × - 2z + 2 × x × - 2z ) - x² - y² - 3z² + 4xy
= (x² + y² + 4z² + 2xy - 4yz - 4xz ) - x² - y² - 3z² + 4xy
= 4z² - 3z² + 2xy + 4xy - 4yz - 4xz
= z² + 6xy - 4yz - 4xz
(iii) Given : (x² – x + 1)² - (x² + x + 1)²
By using an identity (a + b + c)² = a² + b² + c² + 2ab + 2 bc + 2 ca
= ((x²)² + (– x)² + 1² + 2 x² × -x + 2 × -x × 1 + 2 × 1 × x² ) - ((x²)² + ( x)² + 1² + 2 x² × x + 2 × x × 1 + 2 × 1 × x² )
= (x⁴ + x² + 1 - 2x³ - 2 x + 2x² ) - (x⁴ + x² + 1 + 2x³ + 2 x + 2x²)
= (x⁴ + x² + 1 - 2x³ - 2x + 2x² - x⁴ - x² - 1 - 2x³ - 2x - 2x²)
= - 2x³ - 2x³ - 2x - 2x
= - 4x³ - 4x
HOPE THIS ANSWER WILL HELP YOU……
Some more questions :
Find the value of 4x²+y²+25z²+4xy-10yz-20zx when x = 4, y = 3 and z = 2.
https://brainly.in/question/15897597
If a²+b²+c² =16, and ab + bc + ca=10, find the value of a+b+c.
https://brainly.in/question/15897602