Question

In the process of solving 2x^3 + 7x^2 + 9x + 10 = 0, you tested 1, 2, 5, and 10 as possible zeros and determined that none of them are actual zeros. You then discovered that -5/2 is a zero. Do you need to test 1, 2, 5, and 10 again? Why or why not?

Answer 1

No, you do not need to test 1, 2, 5, and 10 again as zeros after finding that -5/2 is a zero, since the polynomial will be reduced to a different equation once -5/2 is factored out.

Step-by-step explanation:

After discovering that -5/2 is a zero of the polynomial equation 2x^3 + 7x^2 + 9x + 10 = 0, you do not need to test potential zeros 1, 2, 5, and 10 again. This is because polynomial equations of degree n will have exactly n roots (real and/or complex), counting multiplicity. Once a zero is found, polynomial division should be applied to simplify the equation by factoring out the zero in the form of a binomial (x - zero). This results in a polynomial of one less degree. When -5/2 is factored out, it will leave you with a quadratic equation, which can be solved further for its two roots using the quadratic formula or factoring, if possible.

Testing potential zeros again is not only redundant but also not mathematically relevant, as you're solving a different, reduced polynomial after factoring out the known zero.