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

By BYJU'S Exam Prep

Updated on: October 17th, 2023

Find the zeroes of the polynomial x2 + x/6 -2 and verify the relation between the coefficient and zeroes of the polynomial

Here are the steps which can be used to find the zeros of the given polynomial:

Step 1: Start with the given polynomial: x2 + x/6 – 2.

Step 2: Set the polynomial equal to zero: x2 + x/6 – 2 = 0.

Step 3: Multiply the entire equation by 6 to eliminate the fraction: 6(x2 + x/6 – 2) = 6(0).

Step 4:Simplify the equation: 6x2 + x – 12 = 0.

Step 5: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let’s use factoring.

Step 6: Factor the quadratic equation: (2x + 3)(3x – 4) = 0.

## 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.