find all the zeroes of 2x4 - 9x3 + 5x2 + 3x-1 if two of its zeroes are 2+root3 and 2-root3
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To find the zeros of the given polynomial, let's use the concept of polynomial division and factor theorem.
Step 1: Divide the given polynomial by the factors formed by the given zeros.
We are given two zeros: 2+√3 and 2-√3. Since these are irrational numbers, we know that the conjugate of each will also be a zero. Hence, the other two zeros are 2-√3 and 2+√3.
Using the factor theorem, we form two factors as follows:
(x - (2 + √3)) and (x - (2 - √3))
Step 2: Perform polynomial division to determine the quadratic factor.
We divide the given polynomial (2x^4 - 9x^3 + 5x^2 + 3x - 1) by the quadratic factor obtained from the conjugate pair of zeros: (x - (2 + √3))(x - (2 - √3)).
Performing this division gives us:
2x^2 - x - 1
Step 3: Find the zeros of the quadratic factor.
Now, we need to solve the quadratic equation 2x^2 - x - 1 = 0 to find the remaining zeros.
Using factoring, we can rewrite the equation as:
(2x + 1)(x - 1) = 0
Setting each factor to zero and solving for x, we get:
2x + 1 = 0 --> x = -1/2
x - 1 = 0 --> x = 1
Therefore, the zeros of the polynomial 2x^4 - 9x^3 + 5x^2 + 3x - 1 are:
x = 2 + √3
x = 2 - √3
x = -1/2
x = 1