# Bionomial Theorem

a) write down the expansion of (3x+5)3 and deduce the expansion for (3x-5)3 b) Hence find the exact solutions to the equation (3x+5)3 - /93x-5)3 = 730

Answers

According to the binomial theorem
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
So (3x+5)^3 = 27x^3 + 135x^2 + 225x + 125
and
(3x-5)^3 = 27x^3 - 135x^2 + 225x - 125

07 February 2011

According to the binomial theorem,
(a+b)^3 = a^3 + 3a^2b + 3ab^2 +b^3
(a) In your question, a=3x and b=5, so
(3x+5)^3 = (3x)^3 + (3((3x)^3)5) + (3(3x)(5^2)) + (5^3)
= 27x^3 +135X^2 + 225X +125
Using the above, we can deduce the solution for (3x-5)^3 as follows:
(3x-5)^3 = (3x)^3 + (3((3x)^3)(-5)) + (3(3x)((-5)^2)) + ((-5)^3)
= 27x^3 -135X^2 + 225X -125
(b) Using the results from part(a), we get that
((3x+5)^3) -((3x-5)^3) = 270x^2 + 250 = 730
270x^2 = 730 - 250
x^2 = (480/270)
= (16/9)
Therefore, x = +/- sqrt(16/9)
= +/- (4/3)
Hence, the exact solutions to the equation are +(4/3) and -(4/3).
Hope this helps.

07 February 2011

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