If alpha and beta are the zeros of the polynomial p(x)=x^2+x+1 then find the value of 1÷alpha+1÷beta 2)alpha^2+beta^2
I will use a for alpha and b for beta
then from the properties of zeros of a quadratic
a+b = -1
ab = 1
then 1/a + 1/b
= (b+a)/(ab) = -1/1 = -1
a^2 + b^2
= (a+b)^2 - 2ab
= (-1)^2 - 2(1)
= 1 - 2 = -1
Thank you very much
Answer
Given polynomial is x2−a(x+1)−b=0
x
2
−ax−a−b=0
x
2
−ax−(a+b)=0
α+β=a,αβ=−a−b.......... given
(α+1)(β+1)=αβ+α+β+1
=αβ+(α+β)+1
=−a−b+a+1=0⇒b=1
1) Well, if alpha and beta are the zeros of the polynomial, that means that plugging alpha or beta into the polynomial will give us 0. So let's try that:
p(alpha) = alpha^2 + alpha + 1 = 0
p(beta) = beta^2 + beta + 1 = 0
Now, let's find 1/alpha + 1/beta:
1/alpha + 1/beta = (alpha + beta) / (alpha * beta)
Hm, seems like we need to find alpha + beta and alpha * beta.
Let's use Vieta's formulas:
alpha + beta = -1
alpha * beta = 1
Plugging these values into the equation:
1/alpha + 1/beta = (-1) / 1
1/alpha + 1/beta = -1
So, the value of 1/alpha + 1/beta is -1.
2) Let's find alpha^2 + beta^2:
alpha^2 + beta^2 = (alpha + beta)^2 - 2alpha*beta
Using Vieta's formulas, we can find:
alpha + beta = -1
alpha * beta = 1
Plugging these values in:
alpha^2 + beta^2 = (-1)^2 - 2(1)
alpha^2 + beta^2 = 1 - 2
alpha^2 + beta^2 = -1
So, the value of alpha^2 + beta^2 is -1.
To find the value of 1/alpha + 1/beta, we need to know the values of alpha and beta.
Given that alpha and beta are the zeros of the polynomial p(x) = x^2 + x + 1, we can use the fact that the sum of the roots of a quadratic polynomial is equal to the coefficient of the linear term (x) with the opposite sign.
In this case, the sum of the roots (alpha + beta) will be equal to the coefficient of x, which is 1. So, we have alpha + beta = -1.
Next, we want to find the product of the roots (alpha * beta). For a quadratic polynomial, the product of the roots is equal to the constant term divided by the coefficient of the quadratic term (x^2).
In this case, the constant term is 1 and the coefficient of the quadratic term is 1, so the product of the roots (alpha * beta) = 1/1 = 1.
Now, we can find the value of 1/alpha + 1/beta.
Using the formula for the sum of fractions, we can rewrite it as (beta + alpha) / (alpha * beta).
Plugging in the values we found earlier, we have (-1) / (1) = -1.
Therefore, the value of 1/alpha + 1/beta is -1.
To find the value of alpha^2 + beta^2, we can use the identity (alpha + beta)^2 - 2(alpha * beta).
Plugging in the values we found earlier, we have (-1)^2 - 2(1) = 1 - 2 = -1.
Therefore, the value of alpha^2 + beta^2 is -1.