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.
To find the values of a, b, and c, we can use the given information that 1 + √3 is one of the zeros of the polynomial and p(2) = -2.
Since 1 + √3 is a zero, we know that (x - (1 + √3)) is a factor of the polynomial. Similarly, the other zero can be found by taking the conjugate of 1 + √3, which is 1 - √3, making (x - (1 - √3)) a second factor.
So, we can express the polynomial p(x) as the product of these two factors:
p(x) = (x - (1 + √3))(x - (1 - √3))
Expanding this expression gives us:
p(x) = (x - 1 - √3)(x - 1 + √3)
Now, let's simplify this expression:
p(x) = ((x - 1) - √3)((x - 1) + √3)
Using the difference of squares identity, we can further simplify:
p(x) = (x - 1)^2 - (√3)^2
p(x) = (x - 1)^2 - 3
To find the values of a, b, and c, we need to write the polynomial in standard form, which is in the form of ax^2 + bx + c.
Comparing p(x) = (x - 1)^2 - 3 with the standard form, we can see that a = 1, b = -2, and c = -3.
Therefore, the values of a, b, and c are a = 1, b = -2, and c = -3.