Final answer:
To find the complex roots of the polynomial 32x² + 12x − 57, the quadratic formula is applied revealing that the roots are complex. The closest option provided by the question is option b) −2 + 3i, −2 − 3i, which matches the expectation that the real part should be negative due to the signs of the coefficients.
Step-by-step explanation:
We need to find the complex roots of the polynomial 32x² + 12x − 57. To do this, we can use the quadratic formula:
x = −(b ± √(b² − 4ac)) / (2a)
Where a, b, and c are coefficients from the polynomial ax² + bx + c. Here, a = 32, b = 12, and c = −57.
Substitute the given values into the quadratic formula:
x = −(12 ± √((12)² − 4 × 32 × (−57))) / (2 × 32)
This simplifies to:
x = −(12 ± √(144 + 7296)) / 64
x = −(12 ± √7440) / 64
Since 7440 is not a perfect square, we will have two complex roots when taking both the positive and negative square roots.
Therefore, the correct answer which represents the complex roots of the polynomial is 'd) −4 + 5i, −4 − 5i', assuming a typo in the provided probabilities since none of them match exactly to our calculated roots. However, if we must choose from the provided options as they are, the logic follows that since the coefficient of x is positive and the constant term is negative, the real part of the complex roots will be negative. Hence, option 'b' is preferred.