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, we need to use the given information that 1 + √3 is one of the zeros of the polynomial and p(2) = -2.
Let's start by finding the other zero of the polynomial. We know that if 1 + √3 is a zero, then its conjugate 1 - √3 must also be a zero. This is because a polynomial with rational coefficients has roots that come in conjugate pairs.
Using these two zeros, we can write the polynomial in factored form:
p(x) = a(x - (1 + √3))(x - (1 - √3))
Expanding this equation, we get:
p(x) = a(x - 1 - √3)(x - 1 + √3)
Simplifying further:
p(x) = a(x^2 - (1 + √3)x - (1 - √3)x + (1 + √3)(1 - √3))
p(x) = a(x^2 - 1√3x - 1√3x + (1 - 3))
p(x) = a(x^2 - 2√3x + 1 - 3)
p(x) = a(x^2 - 2√3x - 2)
Now, we can substitute p(2) = -2 into this expression to solve for a:
-2 = a(2^2 - 2√3(2) - 2)
-2 = a(4 - 4√3 - 2)
-2 = a(2 - 4√3)
-2 = 2a - 4a√3
-2 - 2a = -4a√3
Squaring both sides of the equation to eliminate the square root:
(-2 - 2a)^2 = (-4a√3)^2
4 + 8a + 4a^2 = 16a^2 * 3
4 + 8a + 4a^2 = 48a^2
Rearranging the equation:
48a^2 - 4a^2 - 8a - 4 = 0
44a^2 - 8a - 4 = 0
11a^2 - 2a - 1 = 0
We can now solve this equation for a using the quadratic formula:
a = (-(-2) ± √((-2)^2 - 4(11)(-1))) / (2(11))
Simplifying:
a = (2 ± √(4 + 44)) / 22
a = (2 ± √48) / 22
a = (2 ± 4√3) / 22
a = (1 ± 2√3) / 11
We have two possible solutions for a: a = (1 + 2√3) / 11 and a = (1 - 2√3) / 11.
Now that we have the value of a, we can substitute it back into the factored form of the polynomial to find b and c. Let's consider the solution a = (1 + 2√3) / 11:
p(x) = a(x^2 - 2√3x - 2)
p(x) = ((1 + 2√3) / 11)(x^2 - 2√3x - 2)
Comparing this with p(x) = ax^2 + bx + c, we can see that b = -2√3a / 11 and c = -2a / 11.
Substituting the value of a = (1 + 2√3) / 11:
b = -2√3((1 + 2√3) / 11) / 11
b = (-2√3 - 4 * 3) / 11
b = (-6√3 - 12) / 11
For c:
c = -2((1 + 2√3) / 11) / 11
c = (-2 - 4√3) / 11
Therefore, the values of a, b, and c, when a = (1 + 2√3) / 11, are:
a = (1 + 2√3) / 11
b = (-6√3 - 12) / 11
c = (-2 - 4√3) / 11
Note that since we have two possible solutions for a (the other one being a = (1 - 2√3) / 11), we also have two corresponding sets of values for b and c.