 Sometimes a polynomial has a repeated factor. Remember, when you see an exponent, that means that you have that many factors of whatever the base is. So here this exponent of 2 says that x minus 4 appears as a factor twice, while this polynomial has the factor x squared plus 3x plus 8 five times. The multiplicity of a root is the number of times the corresponding factor appears in a polynomial. For example, we might try to find the roots and their multiplicities. So remember, the roots can be found by setting each factor equal to 0 and solving. So the root x equals 3 came from the factor x minus 3, and this factor appears twice, so the multiplicity is 2. The root x equals negative 8 came from the factor x plus 8. This factor appears 3 times, so the multiplicity is 3, and the roots x equals plus or minus i came from the factor x squared plus 1 equal to 0. This factor only appears once, so the multiplicity of these roots is 1. Or let's consider a polynomial like this. So the roots come from the factors, so one of these factors gives us, and since the factor x squared plus 3x plus 1 appears 3 times, then these roots have multiplicity 3. The other factor gives us some more roots, so we solve, and since this factor appears 5 times, these roots have multiplicity 5. And again, we can also incorporate the multiplicity into the roots, so let's try and write a polynomial with roots x equal 3 with multiplicity 2, and x equals negative 1 with multiplicity 7. So remember, if x equals a is a root, then x minus a is a factor. x equals 3 with multiplicity 2 corresponds to two factors of x minus 3, so we can include them as x minus 3 to the second power. x equals negative 1 with multiplicity 7 corresponds to 7 factors of x plus 1. So we include the factor x plus 1 to the seventh power, and we can even include complex roots. So remember, if a polynomial has real coefficients, complex roots have to appear in conjugate pairs. So the polynomial has to have roots of 1 plus i and also 1 minus i, and so it must have a factor of, and since these roots have multiplicity 2, then this factor must appear twice, and so one such polynomial will be...