Advertisements
Advertisements
Question
Prove that (a + b + c)3 – a3 – b3 – c3 = 3(a + b)(b + c)(c + a).
Advertisements
Solution
To prove: (a + b + c)3 – a3 – b3 – c3 = 3(a + b)(b + c)(c + a)
L.H.S = [(a + b + c)3 – a3] – (b3 + c3)
= (a + b + c – a)[(a + b + c)2 + a2 + a(a + b + c)] – [(b + c)(b2 + c2 – bc)] ...[Using identity, a3 + b3 = (a + b)(a2 + b2 – ab) and a3 – b3 = (a – b)(a2 + b2 + ab)]
= (b + c)[a2 + b2 + c2 + 2ab + 2bc + 2ca + a2 + a2 + ab + ac] – (b + c)(b2 + c2 – bc)
= (b + c)[b2 + c2 + 3a2 + 3ab + 3ac – b2 – c2 + 3bc]
= (b + c)[3(a2 + ab + ac + bc)]
= 3(b + c)[a(a + b) + c(a + b)]
= 3(b + c)[(a + c)(a + b)]
= 3(a + b)(b + c)(c + a) = R.H.S
Hence proved.
APPEARS IN
RELATED QUESTIONS
Factorise the following using appropriate identity:
9x2 + 6xy + y2
Factorise:
`2x^2 + y^2 + 8z^2 - 2sqrt2xy + 4sqrt2yz - 8xz`
Verify that `x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`
Without actually calculating the cubes, find the value of the following:
(–12)3 + (7)3 + (5)3
Without actually calculating the cubes, find the value of the following:
(28)3 + (–15)3 + (–13)3
Simplify of the following:
If `x^4 + 1/x^4 = 194, "find" x^3 + 1/x^3`
If x = 3 and y = − 1, find the values of the following using in identify:
\[\left( \frac{x}{7} + \frac{y}{3} \right) \left( \frac{x^2}{49} + \frac{y^2}{9} - \frac{xy}{21} \right)\]
Find the following product:
(2ab − 3b − 2c) (4a2 + 9b2 +4c2 + 6 ab − 6 bc + 4ca)
If a − b = 5 and ab = 12, find the value of a2 + b2
Mark the correct alternative in each of the following:
If \[x + \frac{1}{x} = 5\] then \[x^2 + \frac{1}{x^2} = \]
If a − b = −8 and ab = −12, then a3 − b3 =
Use the direct method to evaluate the following products:
(x + 8)(x + 3)
Use the direct method to evaluate :
(3b−1) (3b+1)
If `"a" + 1/"a" = 6;`find `"a"^2 - 1/"a"^2`
If p + q = 8 and p - q = 4, find:
p2 + q2
If m - n = 0.9 and mn = 0.36, find:
m2 - n2.
Expand the following:
(3a – 5b – c)2
Expand the following:
(3a – 2b)3
Simplify (2x – 5y)3 – (2x + 5y)3.
