Question

suppose that R(x) is a polynomial of degree 8 whose coefficients are real numbers. also, suppose that r(x) has the following zeros. answer the following.edit: if possible please double check the answers.

Answer 1

Step 1:

In thsi question, we are given that R(x) is a polynomial of degree 8 whose coefficients are real numbers.

Also, suppose that R(x) has the following zeros: 2 , -3 , 5 , - 4 -2i

Step 2 :

For the first sub-question, we are asked to find another zero of R (x) :

From a clear observation, we can see that if -4 - 2i is one of the zeros ,

then the conjugate of -4 - 2i must be another zero,

Hence another zero of R(x ) = -4 + 2 i.

Step 3 :

For the second sub-question, we are asked to find the maximum number of real zeros that R( x) can have :

From the The fundamental theorem of algebra states that

every non-constant single-variable polynomial with complex coefficients has at least one complex root.

This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Hence, there are a maximum of 6 real zeros that R(x) can have.

Step 4 :

From the third sub-question, we are asked to find