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anemone
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Let $a,\,b,\,c$ be the three roots of the equation $8x^3+1001x+2008=0$.
Find $(a+b)^3+(b+c)^3+(c+a)^3$.
Find $(a+b)^3+(b+c)^3+(c+a)^3$.
kaliprasad said:We have from vieta's formula
$a+b+ c = 0\cdots(1)$
$abc = - \frac{2008}{8}\cdots(2)$
hence a+b =-c
b+c = -a
c + a = -b
Hence $(a+b)^3 + (b+c)^3 + (c+a)^3 = -c^3-a^3-b^3 = - (c^3+a^3+b^3)\cdots(3)$
as $a+b+c=0$ so $a^3+b^+c^3 = 3abc \cdots(4)$
from (3) and (4)
$(a+b)^3 + (b+c)^3 + (c+a)^3 = - 3abc = -3 (-\frac{2008}{8})= -753 $ (using (2)
Hence Ans = - 753
The purpose of finding (a+b)³+(b+c)³+(c+a)³ is to simplify and expand a mathematical expression involving binomials raised to the third power. This can help in solving equations or identifying patterns in the expression.
To expand (a+b)³+(b+c)³+(c+a)³, we can use the binomial theorem or the distributive property. First, we expand each binomial raised to the third power, then we combine like terms to simplify the expression.
Yes, (a+b)³+(b+c)³+(c+a)³ can be simplified further by combining like terms. This will result in a final expression with only three terms, instead of six, making it more manageable to work with.
The expansion of (a+b)³+(b+c)³+(c+a)³ can be used in various fields such as physics, engineering, and economics. For example, it can be used to calculate the volume of a cube, to find the total resistance in an electrical circuit, or to determine the total cost of production in a manufacturing process.
Yes, there is a formula for finding (a+b)³+(b+c)³+(c+a)³, known as the "sum of cubes" formula. It states that (a+b)³+(b+c)³+(c+a)³ = 3(a+b)(b+c)(c+a). This formula can save time and effort when expanding and simplifying the expression.