Let be a polynomial in . Prove that if has a root , then has a linear factor in .
Suppose has a root , then is reducible, and then for some nonconstant polynomial . So is a linear factor in .
Is this correct? Did I miss anything important in this proof?
Suppose has a root . By the division algorithm, for some polynomial and for some . Since , we have . So and so is a linear factor of . Is this correct?
Also, for the converse to be true, if has a linear factor, then the linear factor must be a monic polynomial to guarantee that has a root in . Right? That's the only condition I can think of.