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If a,b,c are the roots of x3 − x2 − 1 = 0, what's the value of a/ac + b/ca + c/ab ?
- a)-1
- b)1
- c)2
- d)-2
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If a,b,c are the roots of x3 − x2 − 1 = 0, what's the value of a/ac +...
Under the precondition you can write
(x − a)(x − b)(x − c) = 0 = x3 − x2 − 1
Expanding the product on the left gives
x3 −(a + b + c)x2 + (ab + ac + bc)x − abc = x3 − x2 − 1
Now you have to compare/equate the coefficients on both sides of (∗) and get
a + b + c = 1, ab + ac + bc = 0, abc = 1
Using these and the identity (a + b + c)2 = a2 + b2 + c2 + 2(ab + ac + bc) for evaluation you
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If a,b,c are the roots of x3 − x2 − 1 = 0, what's the value of a/ac +...
To find the value of a/ac, b/ca, and c/ab, we need to find the values of a, b, and c first.
Given that x3 − x2 − 1 = 0, we can rewrite the equation as:
x3 = x2 + 1
This equation tells us that the sum of the roots is equal to the coefficient of x^2, which is 1. So, a + b + c = 1.
The product of the roots can be found by multiplying the constant term of the equation (which is -1) by the coefficient of x^3 (which is 1). Therefore, abc = -1.
Now, let's calculate the value of a/ac:
a/ac = 1/c
We know that a + b + c = 1, so c = 1 - (a + b).
Substituting this value of c in the equation a/ac = 1/c, we get:
a/(1 - a - b) = 1/(1 - (a + b))
Cross-multiplying, we get:
a = (1 - (a + b))
Simplifying further, we get:
2a + b = 1
Similarly, we can calculate the values of b and c:
b = (1 - (a + c))
= (1 - (a + (1 - a - b)))
= (1 - 1)
= 0
c = (1 - (a + b))
= (1 - (a + 0))
= (1 - a)
Now, let's calculate the value of b/ca:
b/ca = (0)/(a(1 - a))
= 0
Finally, let's calculate the value of c/ab:
c/ab = (1 - a)/(a * 0)
= undefined
Therefore, the value of a/ac is 1, the value of b/ca is 0, and the value of c/ab is undefined.
Hence, the correct answer is option B) 1.
Given that x3 − x2 − 1 = 0, we can rewrite the equation as:
x3 = x2 + 1
This equation tells us that the sum of the roots is equal to the coefficient of x^2, which is 1. So, a + b + c = 1.
The product of the roots can be found by multiplying the constant term of the equation (which is -1) by the coefficient of x^3 (which is 1). Therefore, abc = -1.
Now, let's calculate the value of a/ac:
a/ac = 1/c
We know that a + b + c = 1, so c = 1 - (a + b).
Substituting this value of c in the equation a/ac = 1/c, we get:
a/(1 - a - b) = 1/(1 - (a + b))
Cross-multiplying, we get:
a = (1 - (a + b))
Simplifying further, we get:
2a + b = 1
Similarly, we can calculate the values of b and c:
b = (1 - (a + c))
= (1 - (a + (1 - a - b)))
= (1 - 1)
= 0
c = (1 - (a + b))
= (1 - (a + 0))
= (1 - a)
Now, let's calculate the value of b/ca:
b/ca = (0)/(a(1 - a))
= 0
Finally, let's calculate the value of c/ab:
c/ab = (1 - a)/(a * 0)
= undefined
Therefore, the value of a/ac is 1, the value of b/ca is 0, and the value of c/ab is undefined.
Hence, the correct answer is option B) 1.
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If a,b,c are the roots of x3 − x2 − 1 = 0, what's the value of a/ac + b/ca + c/ab ?a)-1b)1c)2d)-2Correct answer is option 'B'. Can you explain this answer?
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