If a and b are the Zeros of the Polynomial f (X)=x² - 5x +k such that a-b =1, Find the Value of k?

If a and b are the Zeros of the Polynomial f(X)=x² - 5x +k such that a-b =1, Find the Value of k?

Solution:

Given that a and b are the zeros of the polynomial f(x) = x² - 5x + k, and a - b = 1, we need to find the value of k.

The sum of the zeros of the polynomial is given by the formula:

Sum of zeros = -(coefficient of x) / (coefficient of x²)

In this case, the sum of the zeros is a + b = 5 / 1 = 5.

We are also given that a - b = 1.

Using these two equations, we can set up a system of equations to solve for a and b.

Equation 1: a + b = 5

Equation 2: a - b = 1

Adding equation 1 and equation 2, we get: (a + b) + (a - b) = 5 + 1

2a = 6

a = 3

Substituting the value of a into equation 2, we have: 3 - b = 1 b = 2

Now that we have the values of a and b, we can substitute them into the original polynomial f(x) = x² - 5x + k and set it equal to zero to find the value of k.

f(x) = x² - 5x + k

Since a and b are the zeros of the polynomial, we have: f(a) = 0 f(b) = 0

Substituting the values of a and b, we get:

(3)² - 5(3) + k = 0

9 - 15 + k = 0

-6 + k = 0

k = 6

Therefore, the value of k is 6.

In summary, if a and b are the zeros of the polynomial f(x) = x² - 5x + k such that a - b = 1, then the value of k is 6.

Answer:

If a and b are the Zeros of the Polynomial f(X)=x² - 5x +k such that a-b =1 , Then the Value of k is 6

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