Final answer:
1/2 and − 3/2 are confirmed as zeros of the polynomial 4x² + 4x − 3 by substituting them into the equation and getting zero as a result. Moreover, the sum and product of the zeros match the negative coefficient of x divided by the coefficient of x² and the constant term divided by the coefficient of x², respectively, confirming the relationship between zeros and coefficients.
Step-by-step explanation:
To show that 1/2 and − 3/2 are the zeros of the polynomial 4x² + 4x − 3, we need to plug these values into the polynomial and see if the result equals zero.
For x = 1/2:
- 4(1/2)² + 4(1/2) − 3 = 4(1/4) + 2 − 3 = 1 + 2 − 3 = 0
For x = − 3/2:
- 4(− 3/2)² + 4(− 3/2) − 3 = 4(9/4) − 6 − 3 = 9 − 6 − 3 = 0
To verify the relationship between zeros and coefficients:
The sum of the zeros (1/2) + (− 3/2) = − 1, which should be equal to −(b/a), while the product of the zeros (1/2) × (− 3/2) = − 3/4, which should be c/a. Substituting our coefficients we obtain:
- −(b/a) = −(4/4) = − 1
- c/a = (− 3)/4 = − 3/4
Both relationships hold true, thus confirming the relationship between zeros and coefficients of the polynomial.