if the zeroes of the polynomial x2-5/2x+k are reciprocal of each other, find the value of k ?
Well, let's give this problem a whirl! If the zeroes of the polynomial are reciprocal of each other, that means if one zero is a, then the other zero is 1/a. So, in this case, let's assume that one zero is a, and the other zero is 1/a.
Now, let's use Vieta's formulas to find the sum and product of the zeroes. The sum of the zeroes is given by the formula: sum_zeros = -b/a = -(5/2), where b represents the coefficient of x. Since we have assumed one zero is a, the other zero is 1/a, and the sum of the zeroes is a + 1/a.
So, we have a + 1/a = -(5/2).
Now, let's find the product of the zeroes using the formula: product_zeros = c/a = k, where c represents the constant term. Since we have assumed one zero is a, the other zero is 1/a, and the product of the zeroes is a * 1/a = 1.
So, we have 1 = k.
Therefore, the value of k is 1.
I hope this explanation didn't make you flip out! 😄
If the zeroes of the polynomial x^2 - (5/2)x + k are reciprocals of each other, then let's assume one of the zeroes to be 'a'.
Recall that if a zero is reciprocal, then its reciprocal is also a zero. Therefore, the other zero can be represented as 1/a.
Now, using the sum and product of the roots, we know that the sum of the roots is equal to -b/a, where 'b' is the coefficient of x, and the product of the roots is equal to c/a, where 'c' is the constant term.
In this case, the sum of the roots is -(-5/2) = 5/2, and the product of the roots is k.
So, we have the equation:
a + (1/a) = 5/2
To solve this equation, let's multiply through by 'a' to eliminate the denominators:
a^2 + 1 = (5/2)a
Now, multiply through by 2 to get rid of the fraction:
2a^2 + 2 = 5a
Rearrange the equation:
2a^2 - 5a + 2 = 0
Now, we can factor this quadratic equation:
(2a - 1)(a - 2) = 0
Setting each factor equal to zero gives:
2a - 1 = 0 or a - 2 = 0
Solving these equations gives:
a = 1/2 or a = 2
Since the roots are reciprocal, the possible pairs of roots are (1/2, 2) and (2, 1/2).
Now, substitute the value of 'k' using the product of the roots:
For (1/2, 2):
k = (1/2)(2) = 1
For (2, 1/2):
k = (2)(1/2) = 1
Therefore, the value of k is 1, regardless of which pair of reciprocal roots is considered.
To find the value of k, we need to first understand the concept of reciprocal zeroes of a polynomial.
Reciprocal zeroes of a polynomial mean that if one zero is denoted by p, then its reciprocal (1/p) is also a zero of the polynomial. In other words, if p is a zero of the polynomial, then (1/p) is also a zero.
Now, let's solve the problem using this concept.
The given polynomial is x^2 - (5/2)x + k.
Let the zeroes of the polynomial be p and (1/p) since they are reciprocals of each other.
According to the Zeroes of a Polynomial Theorem, the sum of the zeroes of a quadratic polynomial is equal to the negative coefficient of x divided by the coefficient of x^2.
So, the sum of the zeroes is p + (1/p) = (5/2)
To simplify this equation, let's clear the denominator by multiplying both sides by 2p:
2p * (p + (1/p)) = 2p * (5/2)
2p^2 + 2 = 5p
Bringing all terms to one side:
2p^2 - 5p + 2 = 0
Now, we need to factorize this quadratic equation:
(2p - 1)(p - 2) = 0
Setting each factor equal to zero:
2p - 1 = 0 or p - 2 = 0
Solving these equations separately:
2p = 1 or p = 2
p = 1/2 or p = 2
Since p cannot be equal to 2 (otherwise, (1/p) would be undefined), we conclude that p = 1/2.
Now, we can find the value of k by substituting p = 1/2 back into the original polynomial:
k = (1/2)^2 - (5/2)(1/2) + k
k = 1/4 - 5/4 + k
1/4 - 5/4 = -4/4 = -1
k - 1 = -1
k = -1 + 1
k = 0
Therefore, the value of k is 0.
So, the answer is k = 0.
In guadratic equation a x² + b x + c = 0
Product of roots = c / a
If your expression mean:
x² - ( 5 / 2 ) x + k = 0 then
In this equqation:
a = 1 , b = - 5 / 2 , c = k
In quadratic eguation product of roots = c / a = k / 1 = k
Given that the roots are reciprocals of each other.
If one root is p, the other would be 1 / p their product will be 1.
So:
Product of roots:
p ∙ 1 / p = 1
Product of roots also:
c / a = k
So k = 1
By the way the soutions of:
x² - ( 5 / 2 ) x + 1 = 0
are
x = 2 and x = 1 / 2