 So, continuing our discussion of the root and remainder theorems, let's see how we can use the fundamental theorem of algebra to find a solution to a polynomial equation. So, suppose x is a polynomial with rational coefficients. The fundamental theorem of algebra, version 1.0, tells us that an n-th degree polynomial can be written as a product of n linear factors. So let's write those out. Now, to keep this as general as possible, we'll write those linear factors in the form something x minus something else. So how can we find the root? Well, a root is any value that makes our expression equal to zero. But because this is a product, this expression will be zero if any of its factors are zero. So the first factor gives us the root factor equal to zero, solving, which gives us one root. The second factor could be zero, so setting that equal to zero on solving gives us... which gives us a second root. For the third factor, we'll do something completely different. No, we won't. The third factor will give us a root by setting it equal to zero and solving. And if we lather, rinse, repeat, we can find all the roots of this polynomial. As a concrete example, suppose I have this product equal to zero, then my solutions are going to occur where any factor is zero. So either 3x minus 7 is zero, 2x plus 5 is zero, or 8x minus 1 is equal to zero. And solving each of these three equations gives us three roots, seven-thirds, minus five-halves, and one-eighth. Now, a little analysis goes a long way. Suppose we expanded this expression over on the left-hand side. Now, you don't really want to expand it because it'll be something of a nightmare to do so. But let's think about what would happen if we did. When we expand, the highest degree term will be formed by multiplying the highest degree terms in each of the factors. So that's going to be 3x times 2x times 8x, which is going to be 48x cubed. Meanwhile, the constant term is going to be formed by multiplying together the constant terms from each factor. That's minus 7 times 5 times negative 1 or 35. And the thing to notice is that every one of our roots is going to come from a factor of 35 divided by a factor of 48. So for example, this root seven-thirds, seven is a factor of 35, three is a factor of 48. And what this suggests is known as the rational root theorem. Suppose capital X is a polynomial with integer coefficients, where the leading coefficient is an and the constant coefficient is a0. Then the rational roots of x, whatever they happen to be, will be of the form x equals plus or minus d0 over dn, where dn is a divisor of an and d0 is a divisor of a0. You can think of these as divisor of the constant over divisor of the leading coefficient. For example, let's try to find the possible rational roots of x cubed minus 12x squared plus 5x plus 12. So the rational root theorem guarantees that if x is a polynomial with integer coefficients, got it, then our rational roots will be of the form plus or minus a divisor of our constant over a divisor of our leading coefficient. And so this means that our roots will be of the form plus or minus a divisor of 12 over a divisor of 1. Well, the only divisor of 1 is 1, so our denominator has to be 1. Our numerator could be a divisor of 12 and so the divisors of 12 are, and so that means our possible roots are plus or minus 1 over 1, plus or minus 2 over 1, plus or minus 3 over 1, 4 over 1, 6 over 1, and 12 over 1, or we might simply just reduce that plus or minus 1, plus or minus 2, 3, 4, 6, or 12. One important thing to remember is that these are the possible roots. The rational root theorem only gives possibilities for the roots. You still have to check to see which, if any, are the actual roots. How about the possibilities for the rational roots of this polynomial? So our rational root theorem applies when we have a polynomial with integer coefficients. Check. And it guarantees that the possible rational roots will have the form a divisor of the constant over a divisor of the leading coefficient. So in this case, the possible rational roots will have the form divisor of 15 over divisor of 2. So the divisors of 15 are the divisors of 2 are, and let's try to find these possibilities systematically. I'll take my divisors of 15 over 1, and this gives me the possibilities. I can also take those divisors of 15 over 2, and that gives me the possibilities. And again, it's vitally important to remember is that these are only possibilities. There are no guarantees in life, and we still don't know what the roots are, but at least we know where to look. So now let's try to find the roots.