<u><em>Explanation:</em></u> <u>The fundamental theorem of Algebra states that:</u> "A polynomial of degree 'n' will have exactly 'n' number of roots"
We know that the degree of the polynomial is given by the highest power of the polynomial. Applying the above theorem on the given question, we can deduce that the polynomial that has exactly 8 roots is the polynomial of the 8th degree
<u>Now, let's check the choices:</u> <u>A. (3x²-4x-5)(2x⁶-5)</u> The term with the highest power will be (3x²)(2x⁶) = 6x⁸ Therefore, the polynomial is of 8th degree which means it has exactly 8 roots. This option is correct.
<u>B. (3x⁴+2x)⁴</u> The term with the highest power will be (3x⁴)⁴ = 81x¹⁶ Therefore, the polynomial is of 16th degree which means it has exactly 16 roots. This option is incorrect.
<u>C. (4x²-7)³</u> The term with the highest power will be (4x²)³ = 64x⁶ Therefore, the polynomial is of 6th degree which means that it has exactly 6 roots. This option is incorrect
<u>D. (6x⁸-4x⁵-1)(3x²-4)</u> The term with the highest power will be (6x⁸)(3x²) = 18x¹⁰ Therefore, the polynomial is of 10th degree which means that it has exactly 10 roots. This option is incorrect
<span>The number of roots of a polynomial is determined by the highest power of the variable after all simplifications has been done.
For the first option, after the brackets has been expanded, the highest power of x will be 8.
The second option will have the highest power of x as 16. The third option will have the highest power of x as 6 while the last option will have the highest power of x as 10.
Therefore, the polynomial function with exactly 8 roots is f(x) = (3x^2 - 4x - 5)(2x^6 - 5).</span>
The shorter period loan has the same total interest for the same loan amount, so it has the higher effective interest rate: 80/12 is higher than 89/14.
