alpha beta gamma are zeros of cubic polynomial x^3-12x^2+44x+c. if alpha beta gamma are in Ap find the value of c class 10
(x-alpha) (x-beta)(x-gama) = x^3 - 12 x^2 + 44x+c. So x^3 - x^2 (alpha+beta+gama)+ a(alphabeta+betagamma+gamma alpha)=alpha beta gama = x^3 - 12 x^2+44x+c, then alpha+beta+gama = -12.
Why did the polynomial go to clown college? Because it wanted to find some funny zeros! Now let's solve this math puzzle, shall we?
If alpha, beta, and gamma are in an arithmetic progression (AP), that means that there is a common difference (d) between them. We can express this as:
beta = alpha + d
gamma = alpha + 2d
Since these are the zeros of the cubic polynomial, we know that their sum is equal to the coefficient of the x^2 term divided by the coefficient of the x^3 term. In this case, that's -(-12). So:
alpha + beta + gamma = -(-12)
alpha + alpha + d + alpha + 2d = 12
3alpha + 3d = 12
Now let's use one of the zeros, let's say alpha, to find the value of c. We can do this by substituting alpha=0 into the polynomial equation:
0^3 - 12(0^2) + 44(0) + c = 0
c = 0
Therefore, the value of c is 0. Not much of a clown surprise there, huh?
To find the value of "c" in the cubic polynomial x^3 - 12x^2 + 44x + c, given that the zeros (roots) of the polynomial form an arithmetic progression (AP), we can use a strategy based on Vieta's formulas.
Vieta's formulas state that for a cubic polynomial in the form ax^3 + bx^2 + cx + d = 0 with roots α, β, and γ, the following relationships hold:
α + β + γ = -b/a
αβ + βγ + γα = c/a
αβγ = -d/a
In this case, we are given that α, β, and γ form an arithmetic progression. Let's assume the common difference of the AP is 'd'. This means that β = α + d and γ = α + 2d.
Substituting these values into the first equation (α + β + γ = -b/a):
α + (α + d) + (α + 2d) = -(-12)/1
3α + 3d = 12
α + d = 4
Now, let's substitute these expressions for α and β into the second equation (αβ + βγ + γα = c/a):
(α)(α + d) + (α + d)(α + 2d) + α(α + 2d) = c/1
α^2 + αd + α^2 + 3αd + 2d^2 + α^2 + 2αd = c
3α^2 + 6αd + 2d^2 = c
We can also use the fact that αβγ = -d/1:
α(α + d)(α + 2d) = -d/1
α^3 + 3α^2d + 2αd^2 = -d
Now, let's substitute the value of α + d from the previous equation into this equation:
(4 - d)^3 + 3(4 - d)^2d + 2(4 - d)d^2 = -d
Expanding and simplifying this equation will provide the value of 'd'. After finding 'd', we can substitute it back into the equation α + d = 4 to find the value of α. Finally, we can substitute the values of α, β, and γ into the polynomial to calculate the value of 'c'.
To find the value of c, we need to determine the relationship between the zeros (alpha, beta, and gamma) since they are in an arithmetic progression (AP).
Let's assume that the common difference of the arithmetic progression is d.
According to the properties of an arithmetic progression, we have:
beta = alpha + d
gamma = beta + d
Substituting these values into the given cubic polynomial, we get:
x^3 - 12x^2 + 44x + c = (x - alpha)(x - beta)(x - gamma)
= (x - alpha)(x - (alpha + d))(x - (alpha + 2d))
Expanding this expression, we have:
x^3 - (3alpha + 3d)x^2 + (3alpha^2 + 6alpha*d + d^2)x - (alpha^3 + 3alpha^2*d + 2alpha*d^2)
Comparing the coefficients of the polynomial with the expanded expression, we can equate their corresponding terms:
- 12x^2 = - (3alpha + 3d)x^2 [Coefficient of x^2]
44x = (3alpha^2 + 6alpha*d + d^2)x [Coefficient of x]
c = - (alpha^3 + 3alpha^2*d + 2alpha*d^2) [Constant term]
Now, since we know that alpha, beta, and gamma are the zeros of the cubic polynomial, we can write the following equations:
- (3alpha + 3d)x^2 = - 12x^2 (Equation 1)
(3alpha^2 + 6alpha*d + d^2)x = 44x (Equation 2)
-(alpha^3 + 3alpha^2*d + 2alpha*d^2) = c (Equation 3)
From Equation 1, we get:
3alpha + 3d = 12 (dividing both sides by -x^2 and multiplying by -1)
Simplifying, we have:
alpha + d = 4 (dividing both sides by 3 and multiplying by -1)
From Equation 2, we get:
3alpha^2 + 6alpha*d + d^2 = 44 (dividing both sides by x)
Simplifying, we have:
alpha^2 + 2alpha*d + (d^2/3) = (44/x) (dividing both sides by 3 and multiplying by -1)
From the given information that alpha, beta, and gamma are in an arithmetic progression, we know that beta = alpha + d. So substituting beta = alpha + d in Equation 3, we get:
- (alpha^3 + 3alpha^2*d + 2alpha*d^2) = c
Expanding this equation, we have:
- alpha^3 - 3alpha^2*d - 2alpha*d^2 = c
Now, substituting the value of alpha^2 + 2alpha*d from Equation 2, we get:
- alpha^3 - (alpha^2 + 2alpha*d) - 2alpha*d^2 = c
Simplifying, we have:
- alpha^3 - alpha^2 - 2alpha*d - 2alpha*d^2 = c
From the property of an arithmetic progression, we have the following relationship between alpha, beta, and gamma:
2beta = alpha + gamma (since beta = alpha + d and gamma = beta + d)
Substituting these values, we get:
2(alpha + d) = alpha + (alpha + 2d)
Simplifying, we have:
2alpha + 2d = 2alpha + 2d
This equation holds true, so we can conclude that our assumptions are correct.
Hence, the value of c is - alpha^3 - alpha^2 - 2alpha*d - 2alpha*d^2.
To find the specific value of c, we need the values of alpha and d. Unfortunately, the question does not provide these values. Therefore, we cannot determine the exact value of c without more information.