If alpha,beta,gamma,delta be the roots of the equation x^4-x^3+2x^2+x+1=0,find the value of 1. (2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1)
2. (alpha square + 1)(beta square + 1) (gamma square + 1)(delta square + 1)
Please solve this solution
To find the value of the given expressions, we can use Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to its roots.
Let's start by finding the sum and the product of the roots of the given equation.
The given equation is:
x^4 - x^3 + 2x^2 + x + 1 = 0
The sum of the roots, by Vieta's formulas, is equal to the negation of the coefficient of the x^3 term:
alpha + beta + gamma + delta = 1/1 = 1
The product of the roots is equal to the constant term divided by the coefficient of the x^4 term:
alpha * beta * gamma * delta = 1/1 = 1
Now, let's find the value of the first expression: (2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1)
We can rewrite it as: (2^4)(alpha + 1/2)(beta + 1/2)(gamma + 1/2)(delta + 1/2)
To find this value, we need to find the sum of the roots plus 1/2 for each root:
(2 alpha + 1) = 2(alpha + 1/2)
(2 beta + 1) = 2(beta + 1/2)
(2 gamma + 1) = 2(gamma + 1/2)
(2 delta + 1) = 2(delta + 1/2)
Now, substituting the values for the sum of the roots, we get:
(2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1) = 2^4 * (alpha + 1/2)(beta + 1/2)(gamma + 1/2)(delta + 1/2) = 2^4 * (1 + 1/2)^4 = 16 * (3/2)^4 = 16 * 81/16 = 81
Therefore, the value of (2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1) is 81.
Now, let's find the value of the second expression: (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1)
We can rewrite it as: (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1)
To find this value, we need to find the sum of the squares of the roots plus 1 for each root:
(alpha^2 + 1) = (alpha^2 + 1)
(beta^2 + 1) = (beta^2 + 1)
(gamma^2 + 1) = (gamma^2 + 1)
(delta^2 + 1) = (delta^2 + 1)
Now, substituting the values for the sum of the squares of the roots, we get:
(alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1) = (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1) = (1^2 + 1)(1^2 + 1)(1^2 + 1)(1^2 + 1) = 4 * 4 * 4 * 4 = 256
Therefore, the value of (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1) is 256.
To find the value of (2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1), we can use the relationship between the roots and the coefficients of the polynomial equation.
Given the equation x^4 - x^3 + 2x^2 + x + 1 = 0, we can rewrite it as:
(x - alpha)(x - beta)(x - gamma)(x - delta) = 0
Expanding this equation, we get:
x^4 - (alpha + beta + gamma + delta)x^3 + (alpha beta + alpha gamma + alpha delta + beta gamma + beta delta + gamma delta)x^2 - (alpha beta gamma + alpha beta delta + alpha gamma delta + beta gamma delta)x + alpha beta gamma delta = 0
Comparing the coefficients of the expanded equation with the given polynomial, we can deduce the following relationships:
1. The sum of the roots:
alpha + beta + gamma + delta = 1
2. The sum of the product of all possible pairs of roots (taken two at a time):
alpha beta + alpha gamma + alpha delta + beta gamma + beta delta + gamma delta = 2
3. The sum of the product of all possible triples of roots (taken three at a time):
alpha beta gamma + alpha beta delta + alpha gamma delta + beta gamma delta = 1
4. The product of all four roots:
alpha beta gamma delta = 1
Now we can substitute these relationships into the expression (2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1) to simplify it.
(2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1) =
(2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1) =
(4 alpha beta gamma delta + 2(alpha beta gamma + alpha beta delta + alpha gamma delta + beta gamma delta) + alpha beta + alpha gamma + alpha delta + beta gamma + beta delta + gamma delta + 1) =
(4 + 2(1) + 2) =
8
Therefore, the value of (2 alpha + 1)(2 beta + 1)(2 gamma + 1)(2 delta + 1) is 8.
Now let's move on to the second part of the question.
To find the value of (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1), we will again use the relationships between the roots and the coefficients of the polynomial equation.
Expanding (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1), we get:
(alpha^2 beta^2 gamma^2 delta^2) + (alpha^2 beta^2 gamma^2 + alpha^2 beta^2 delta^2 + alpha^2 gamma^2 delta^2 + beta^2 gamma^2 delta^2) + (alpha^2 beta^2 + alpha^2 gamma^2 + alpha^2 delta^2 + beta^2 gamma^2 + beta^2 delta^2 + gamma^2 delta^2) + (alpha^2 + beta^2 + gamma^2 + delta^2) + 1
Using the relationships we deduced earlier, we can simplify this expression:
(alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1) =
(1 + 1)(2 + 1)(2 + 1)(1 + 1) =
2 * 3 * 3 * 2 =
36
Therefore, the value of (alpha^2 + 1)(beta^2 + 1)(gamma^2 + 1)(delta^2 + 1) is 36.