verify the relationship between the zeroes and the coefficient in the polynomial f (x)=root 4x square +5x-2 root3 if alpha and beta are the zeroes of the polynomial x*2+x-2 . Find the values of 1/alpha -1/beta. Find the value alphacube betasquare + alphasquare betacube
To verify the relationship between the zeroes and the coefficients of a polynomial, we use Vieta's formulas.
For a polynomial of the form f(x) = ax^2 + bx + c, where α and β are the zeroes, Vieta's formulas state:
1. The sum of the zeroes is given by α + β = -b/a.
2. The product of the zeroes is given by α * β = c/a.
Now, let's apply Vieta's formulas to the given polynomial f(x) = √4x^2 + 5x - 2√3:
1. The sum of the zeroes (α + β) is equal to -b/a.
Comparing with our polynomial, we can see that a = 4, b = 5. So, -b/a = -5/4.
2. The product of the zeroes (α * β) is equal to c/a.
Comparing with our polynomial, we can see that a = 4, c = -2√3. So, c/a = -2√3/4 = -√3/2.
Now, moving on to the second part of the question:
We are given the polynomial x^2 + x - 2, and we need to find the value of (1/α) - (1/β).
To find this value, we can use the relationship between the sum and product of the roots:
(1/α) - (1/β) = (β - α) / (α * β)
Using Vieta's formulas for this polynomial, we know that α + β = -1 and α * β = -2.
So, substituting these values into the equation, we have:
(1/α) - (1/β) = (β - α) / (-2)
Now, simplifying further, we get:
(β - α) / (-2) = (-1 - α) / (-2) = (α + 1) / 2
Therefore, the value of (1/α) - (1/β) is (α + 1) / 2.
Finally, let's find the value of α^3 * β^2 + α^2 * β^3:
Using the relationship between the product and sum of the roots again, we have:
α^3 * β^2 + α^2 * β^3 = α^2 * α * β^2 + β^3 * α^2 * β
Simplifying further, we get:
α^2 * α * β^2 + β^3 * α^2 * β = α^3 * β^2 * (α + β) + α^2 * β^3 * (α + β)
Using Vieta's formulas, we know that α + β = -1. Substituting this value, we have:
α^3 * β^2 * (-1) + α^2 * β^3 * (-1) = -α^3 * β^2 - α^2 * β^3
Therefore, the value of α^3 * β^2 + α^2 * β^3 is -α^3 * β^2 - α^2 * β^3.
I hope this explanation helps you understand how to verify the relationship between the zeroes and coefficients of a polynomial, as well as solve the given equations.
To verify the relationship between the zeroes and the coefficients in the polynomial f(x) = √4x^2 + 5x - 2√3, we need to use Vieta's formulas.
Vieta's formulas state that for a quadratic polynomial ax^2 + bx + c = 0, the relationship between the coefficients and the zeroes is as follows:
1. The sum of the zeroes (alpha + beta) is equal to -b/a.
2. The product of the zeroes (alpha * beta) is equal to c/a.
In the given polynomial f(x) = √4x^2 + 5x - 2√3, we can see that it is not in the standard form ax^2 + bx + c = 0. However, we can rewrite it as follows:
√4x^2 + 5x - 2√3 = 0
Now, comparing this with the standard form, we have:
a = √4 = 2
b = 5
c = -2√3
Using the above values, we can apply Vieta's formulas to find the relationship between the zeroes and the coefficients:
1. Sum of the zeroes (alpha + beta):
- (sum of the zeroes) = b/a
- (alpha + beta) = 5/2
2. Product of the zeroes (alpha * beta):
- (product of the zeroes) = c/a
- (alpha * beta) = -2√3/2 = -√3
Now, let's move on to the next part of the question.
Given the zeroes of the polynomial x^2 + x - 2, which are alpha and beta, we need to find the value of 1/alpha - 1/beta.
To find the value of 1/alpha - 1/beta, we need to find the value of (beta - alpha) / (alpha * beta) using Vieta's formulas again.
From Vieta's formulas, we know that:
- (sum of the zeroes) = -b/a
- (product of the zeroes) = c/a
For the polynomial x^2 + x - 2, we can see that:
- (sum of the zeroes) = -b/a = -1
- (product of the zeroes) = c/a = -2
Using the values above, we find:
(beta - alpha) = -1
(alpha * beta) = -2
Therefore, the value of 1/alpha - 1/beta can be calculated as follows:
1/alpha - 1/beta = (beta - alpha) / (alpha * beta) = (-1) / (-2) = 1/2.
Finally, to find the value of alpha^3 * beta^2 + alpha^2 * beta^3, we can substitute the values of alpha and beta into the expression and simplify:
alpha^3 * beta^2 + alpha^2 * beta^3 = (alpha^2 * alpha) * (beta^2) + (alpha^2) * (beta^3)
= alpha^5 * beta^2 + alpha^2 * beta^5
Without knowing the specific values of alpha and beta, we cannot simplify the expression any further.