Find the zeroes of the quadratic polynomial 4x2 - 4x - 3 and verify the relationship between the zeroes and the coefficients of the polynomial.

f(x) = 4x2 - 4x - 3. Find the Zeroes and Verify the Relationship between the Zeroes and the Coefficients.

The question states "Find the zeroes of the quadratic polynomial 4x2-4x-3 and verify the relationship between the zeroes and the coefficients of the polynomial.” The zeros of the following quadratic polynomials 4x2 - 4x - 3 are 3/2 and -½ and the relationship between the zeroes and the coefficients of the polynomial is verified in the below steps. Follow these three steps to find the zeros of the following quadratic polynomials 4x2 - 4x - 3:

Step 1: Form the equation

The given polynomial is 4x2 - 4x - 3

We are aware that a polynomial's zeroes are evaluated by equating them to zero.

p(s) = 0

Hence the above equation becomes 4x2 - 4x - 3 = 0.

Step 2: Solve to find the zeroes

 4x2 - 4x - 3 = 0

The above equation can be written as:

4x2 - 6x + 2x - 3 = 0

2x (2x - 3) + 1 (2x - 3) = 0 

(2x + 1)(2x - 3) = 0

Hence, x = 3/2 and -½

Step 3: Verification

We know that for a given polynomial as2 + bs + c

Sum of the zeroes = -b/a and product of the zeroes = c/a

Substituting the values in the above formula we get:

Sum of the zeroes = 3/2 + -½ = 1

Again, -b/a = - (-4)/4 = 1

Product of the zeroes = 3/2 * -½ = -¾

Again, c/a = -¾

Thus, the relationship between the zeroes and the coefficients of the polynomial 4x2 - 4x - 3 is verified. Hence, the zeroes of this polynomial are 3/2 and -½.

Summary:

The zeros of the quadratic polynomial 4x2 - 4x - 3 are 3/2, -½, and the relationship between the zeroes and the coefficients of the polynomial is verified. By substituting values in the sum of the zeroes = -b/a and the product of the roots = c/a, we can easily find the zeros of the quadratic polynomial 4x2 - 4x - 3.

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