 A useful feature of mathematics is we can often run it backwards. So remember that if x equals a is a root, then x minus a is a factor. And this means we can construct polynomials with specific roots. So for example, let's try to find three different polynomials with roots x equal to 3, and x equals 1 plus square root of 7. So since x equals 3 is a root, then our polynomial has a factor of x minus 3. So our polynomial has to be x minus 3 times maybe something else. But wait, since x equals 1 plus square root of 7 is a root, then the polynomial has a factor of x minus 1 plus square root of 7. And when we expand this polynomial, it will have the roots x equals 3 and x equals 1 plus square root of 7. Now the polynomial could have other factors. For example, we could get another such polynomial by including a factor of, oh I don't know how about x plus 8. Or we could make another polynomial by including a different factor, how about x squared plus 7x minus 17. And as long as we keep the factors x minus 3 and x minus 1 plus square root of 7, the polynomial will have roots of 3 and 1 plus square root of 7 plus possibly some other roots. Now suppose ax squared plus bx plus c is a factor of the polynomial, then we can use the quadratic formula to find the roots. And what's important here is that if b squared minus 4ac is negative, these roots will be complex. But since we're adding and subtracting them, this means that if a polynomial with real coefficients has complex roots, the roots occur in conjugate pairs. Now let's say we want to find and expand a polynomial with real coefficients with a root of 3 minus i. Since 3 minus i is a root, then x minus 3 minus i is a factor. And so one polynomial could be x minus 3 minus i. But this polynomial doesn't have real coefficients. So it's not a polynomial with real coefficients with a root of 3 minus i. Now since the polynomial has real coefficients, the complex roots have to occur in conjugate pairs, and the conjugate of 3 minus i is 3 plus i. And so that means 3 plus i is another root, and so x minus 3 plus i is another factor. Now let's expand this out and verify that we actually get a polynomial with real coefficients. And to expand this, it helps to notice that the form of our roots allows us to rewrite the two factors, which means that our product is of the form a plus b times a minus b, which we know as a squared minus b squared. So when we expand it, we get x minus 3 squared minus i squared. And we can expand and simplify. And as requested, we have a polynomial with real coefficients. And if we have more roots, we'll just include more factors. So here we have roots 3 and 2 plus 3i. And so the factored form of such a polynomial could be expanded. It must include the factor of x minus 3 and x minus 2 plus 3i. But since the complex roots occur in conjugate pairs, then 2 minus 3i is another root, and that gives us another factor. And as before, let's expand the terms that give us the complex roots first. So again, it's helpful to remember that we can rewrite this. So we get x minus 2 minus 3i times x minus 2 plus 3i, again, a minus b times a plus b. That gives us a squared minus b squared, which we can expand and simplify, and then expand once again to get our final expanded polynomial. And sometimes we might not need to find the actual expanded polynomial. We might be interested in knowing things like the least degree of a polynomial, which has roots 2, 5, and 3i. So the polynomial must include the factors of, and you can't leave a complex root by itself. You also have to have the conjugate of that root. So we also have to have a factor of x plus 3i. So while our polynomial could include other factors, it doesn't have to. And since we're looking for the least degree polynomial, this means our polynomial must be of at least the fourth degree.