Final answer:
Substituting 1/2 and -3/2 into the polynomial 4x^2 + 4x - 3 confirms they are zeros. Additionally, the sum of the zeros equals -b/a and their product equals c/a, satisfying the relationship between the zeros and the polynomial's coefficients.
Step-by-step explanation:
To show that 1/2 and -3/2 are the zeros of the polynomial 4x^2 + 4x - 3, we can substitute these values into the polynomial and verify that the result is zero. A zero of a polynomial is a value for x that makes the polynomial equal to zero. For the given polynomial, its general form is ax^2 + bx + c = 0, where a = 4, b = 4, and c = -3.
- Substituting x = 1/2, we get 4(1/2)^2 + 4(1/2) - 3 = 4(1/4) + 4(1/2) - 3 = 1 + 2 - 3 = 0, which confirms that 1/2 is a zero.
- Substituting x = -3/2, we get 4(-3/2)^2 + 4(-3/2) - 3 = 4(9/4) - 6 - 3 = 9 - 6 - 3 = 0, which confirms that -3/2 is a zero.
To verify the relationship between the zeros and the coefficients of the polynomial, we use the fact that sum of the zeros (-b/a) should equal the coefficient of x (4) divided by the coefficient of x^2 (4), and the product of the zeros (c/a) should equal the constant term (-3) divided by the coefficient of x^2 (4).
- Sum of zeros: (1/2) + (-3/2) = -2/2 = -1 which is equal to -b/a = -4/4 = -1.
- Product of zeros: (1/2) * (-3/2) = -3/4 which is equal to c/a = -3/4.
This confirms the relationship between zeros and coefficients.