Find the Zeroes of the Polynomial x 2 + x/6 -2 and Verify the Relation between the Coefficient and Zeroes of the Polynomial

Find the Zeroes of the Polynomial x² + x/6 -2 and Verify the Relation between the Coefficient and Zeroes of the Polynomial

To find the zeroes of the polynomial x² + x/6 - 2, we set the polynomial equal to zero and solve for x:

x² + x/6 - 2 = 0

Multiplying the entire equation by 6 to eliminate the fraction:

6x² + x - 12 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Using factoring:

(2x + 3)(3x - 4) = 0

Setting each factor equal to zero:

2x + 3 = 0 or 3x - 4 = 0

Solving each equation:

2x = -3 --> x = -3/2

3x = 4 --> x = 4/3

So, the zeroes of the polynomial x²+ x/6 - 2 are x = -3/2 and x = 4/3.

Now, let's verify the relation between the coefficients and zeroes of the polynomial. For a quadratic polynomial in the form ax² + bx + c, the sum of the zeroes is equal to -b/a, and the product of the zeroes is equal to c/a.

In this case, the coefficient of x² is 1, the coefficient of x is 1/6, and the constant term is -2.

The sum of the zeroes: (-1/6) / 1 = -1/6

The product of the zeroes: (-2) / 1 = -2

We can observe that the sum of the zeroes (-1/6) matches the negative coefficient of x (-1/6), and the product of the zeroes (-2) matches the constant term (-2). Hence, the relation between the coefficients and zeroes of the polynomial is verified.

Answer

The zeros of the polynomial x²+ x/6 - 2 are x = -3/2 and x = 4/3.

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