If one of the zeroes of the cubic polynomial x3+ax2+bx+c is -1 then the product of the other two zeroes is -------(A) b -a +1 (B) b-a-1 (C) a-b+1 (D) a-b-1
when u divide the cubic polynomial by x+1 u shud get remainder as 0 bcoz given that -1 is a zero..
When u divide the cubic polynomial ul get remainder as c-b+a-1 which is = 0
From here u can say that c=b-a+1
Therefore option numberA is correct
The product of all three roots is -c. So, if one root is -1, the other two must have a product of c.
Hmmm. Not what you wanted. So, divide by x+1, and you have
x^3 + ax^2 + bx + c
= (x+1)(x^2 + (a-1)x + (b-a+1))
So, the other two roots must have a product of b-a+1. Also, of course, you need c-b+a-1 = 0.
Check: Pick a,b,c such that a+c=b+1
For example, a=5,c=3,b=7
x^3+5x^2+7x+3 = (x+1)(x+1)(x+3)
and the other two roots multiply to 3, and 7-5+1 = 3.
To find the product of the other two zeroes of the cubic polynomial given that one of the zeroes is -1, we can use Vieta's formulas.
Vieta's formulas state that for a cubic polynomial in the form of x^3 + ax^2 + bx + c = 0,
The sum of the zeroes is given by: -a
The product of the zeroes is given by: -c
Given that one zero is -1, we can substitute this value into the formula for the product of the zeroes:
Product of the other two zeroes = -c/(-1)
Product of the other two zeroes = c
Therefore, the product of the other two zeroes of the cubic polynomial is c.
Answer: None of the provided options.
To find the product of the other two zeroes of the cubic polynomial, we first need to know the relationship between the zeroes and the coefficients of the polynomial.
Let's consider the polynomial x^3 + ax^2 + bx + c. We are given that one of the zeroes is -1. This means that when we substitute -1 for x in the polynomial, the result should be equal to zero.
Substituting -1 into the polynomial, we get:
(-1)^3 + a(-1)^2 + b(-1) + c = 0
-1 + a - b + c = 0
a - b + c = 1
Now, we can use the relationship between the sum and product of zeroes to find the product of the other two zeroes.
The sum of the zeroes is given by:
sum_of_zeroes = -1 + other_zeroes_1 + other_zeroes_2
Since the sum of the zeroes is equal to the coefficient of the quadratic term divided by the coefficient of the cubic term (a / 1 = a), we have:
sum_of_zeroes = a
Using the relationship between the sum and product of zeroes, we know that:
product_of_zeroes = (cubic term coefficient) / (cubic term coefficient) = c / 1 = c
Therefore, the product of the other two zeroes can be found by subtracting the given zero (-1) and the sum of the zeroes (a) from the product of all three zeroes (c):
product_of_other_zeroes = product_of_zeroes - (-1) - sum_of_zeroes
= c - (-1) - a
= c + 1 - a
So, the product of the other two zeroes is c + 1 - a.
In option (A), the expression is b - a + 1, which is different from our calculated expression c + 1 - a. Therefore, option (A) is not the correct answer.
In option (B), the expression is b - a - 1, which matches our calculated expression c + 1 - a. Therefore, option (B) is the correct answer.
Option (C), a - b + 1, and option (D), a - b - 1, do not match the calculated expression. Therefore, options (C) and (D) are not the correct answers.
Therefore, the correct answer is (B) b - a - 1.