Final answer:
To determine the values of a and b so that the polynomial x⁴ + 8x³ - 5x² + ax + b is divisible by 2x² - 5, we can use the quadratic formula to find the roots of the divisor polynomial and perform polynomial division. The coefficients a and b will emerge as part of this division.
Step-by-step explanation:
Finding Coefficients a and b
To find the values of a and b for the polynomial x⁴ + 8x³ - 5x² + ax + b being exactly divisible by 2x² - 5, we will use the fact that the remainder is zero when the first polynomial is divided by the second polynomial. Since the polynomial is of the second degree, it will have two roots, r1 and r2. These roots can be found using the quadratic formula: -b ± √(b² - 4ac) / (2a). Applying this to the polynomial 2x² - 5, we can find the values of r1 and r2 and then use them in polynomial long division, or synthetic division, to find the coefficients a and b.
First, let's find the roots of 2x² - 5:
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- a = 2
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- b = 0 (since there is no x-term)
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- c = -5
Substitute these values into the quadratic formula:
r1 and r2 = -0 ± √(0) - 4(2)(-5) / (2 * 2)
Now we have two roots of the polynomial 2x² - 5 and we can use these roots in the division process.
The coefficient a will be the term accompanying x after division, and the constant b will be the remainder. After finding a and b, we can then check which of the given options a) to d) match our answer.