A polymomial p(x) = ax^2 + bx + c where a, b, c are some rational, has 1 + root 3 as one of the zeros and also p (2) = - 2, Find values of a, b and c. Please work the solution. Tried it many times but still can't get the answer/ The answer is a = 1, b = -2 and c = -2.
To find the values of a, b, and c in the polynomial equation p(x) = ax^2 + bx + c given the conditions that one of the zeros is 1 + √3 and p(2) = -2, you can follow these steps:
Step 1: Use the fact that one of the zeros is 1 + √3 to find the other zero.
Since the polynomial has rational coefficients, complex conjugate pairs are guaranteed. Therefore, the other zero must be the complex conjugate of 1 + √3. We can find this by taking the conjugate:
Conjugate of 1 + √3 = 1 - √3.
So, the second zero is 1 - √3.
Step 2: Use the zeros to express the quadratic equation.
The quadratic equation with roots 1 + √3 and 1 - √3 can be written as:
(x - (1 + √3))(x - (1 - √3)) = 0
Simplifying this equation, we have:
(x - 1 - √3)(x - 1 + √3) = 0
Step 3: Expand the equation and write it in standard form.
Expanding the above equation, we get:
x^2 - (1 + √3)x - (1 - √3)x + (1 + √3)(1 - √3) = 0
Simplifying further:
x^2 - 2x + (1 - 3) = 0
x^2 - 2x - 2 = 0
Step 4: Compare the expanded form with the given polynomial p(x) = ax^2 + bx + c.
By comparing the coefficients of the expanded form with the given polynomial, we can equate them:
a = 1
b = -2
c = -2
Therefore, the values of a, b, and c are:
a = 1, b = -2, c = -2.
These values satisfy the conditions of the given polynomial p(x) and the given zero and value of p(2).