Expand each of the following, using suitable identities:
() (x+2y + 4z)
(ii) (2x-y+z)?
(iii) (-2x + 3y + 2-)
(iv) (3a - 7b-c)?
(v) (-2x + 5y-32)
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Answers
(i) (x+2y+4z)²
Using : (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz
Here,
x = x , y = 2y , z = 4z
(x+2y+4z)² = (x)² + (2y)² + (4z)² + 2*x*2y + 2*2y*4z + 2*x*4z
= x² + 4y² + 16z² + 4xy + 16yz + 8xz
➖➖➖➖➖➖➖➖➖➖➖
(ii) ( 2x - y+z)²
Using : (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz
Here,
x = 2x , y = (-y) , z = z
( 2x - y+z)² = (2x)² + (-y)² + (z)² + 2*2x*(-y) + 2*(-y) *z + 2*2x*z
= 4x² + y² + z² - 4xy - 2yz + 4xz
➖➖➖➖➖➖➖➖➖➖➖
(iii) (-2x+3y+2z)²
Using : (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz
Here,
x = (-2x) , y = 3y , z = 2z
(-2x+3y+2z)² = (-2x)² + (3y)² + (2z)² + 2*(-2x)*3y + 2*3y*2z + 2*(-2x)*2z
= 4x² + 9y² + 4z² - 12xy + 12yz - 8xz
➖➖➖➖➖➖➖➖➖➖➖
(iv) (3a-7b-c)²
Using : (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz
Here,
x = 3a , y = (-7b) , z = (-c)
(3a-7b-c)² = (3a)² + (-7b)² + (-c)² + 2*3a*(-7b) + 2*(-7b)*(-c) + 2*3a*(-c)
= 9a² + 49b² + c² - 42ab + 14bc - 6ac
➖➖➖➖➖➖➖➖➖➖➖
(v) (-2x+5y-3z)²
Using : (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz
Here,
x = (-2x) , y = 5y , z = (-3z)
(-2x+5y-3z)² = (-2x)² + (5y)² + (-3z)² + 2*(-2x)*5y + 2*5y*(-3z) + 2*(-2x)*(-3z)
= 4x² + 25y² + 4z² - 20xy - 30yz + 12xz
➖➖➖➖➖➖➖➖➖➖➖
(VI) (¼a - ½b + 1)²
Using : (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz
Here,
x = (¼a) , y = (-½b) , z = 1
(¼a - ½b + 1)² = (¼a)² + (-½b)² + (1)² + 2*¼a*(-½b) + 2*(-½b)*1 + 2*(¼a) *1
= 1/16 a² + 1/4 b² + 1 - 1/4 ab - b + 1/2 a
= a²/16 + b²/4 + 1 - (ab)/4 - b + a/2
Answer:
( a + b + c )^2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
Step-by-step explanation:
(i) (x+2y+4z)2 = (x)2 + (2y)2 + (4z)2 + 2×x×2y + 2×2y×4z + 2×x×4z
= x2 + 4y2 + 8z2 + 4xy + 16yz + 8xz
(ii) (2x-y+z)2 = 4x2 + y2 + z2 + -2xy + -2yz + 4xz
(iii) (-2x+3y+2) = 4x2 + 9y2 + 4 + -12xy + 12y + -8x
(iv) (3a-7b-c) = 9a2 + 14b2 + c2 + -42ab + 14bc + -6ac
(v) (-2x+5y-32) = 4x + 25y + 1024 + -20xy + -320y + 128x
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