If factor x-k occurs m times in the factorization of a polynomial, then k is a zero of multiplicity m. A zero of multiplicity 2 is called a double zero. For a polynomial, the number of zeros will be less than or equal to its degree. The sum of all the multiplicities of a polynomial will be equal to its degree. f(x) = (x-1)(x-3)(x-3) is a third degree polynomial with 2 zeros, where (x-1) is a zero with a multiplicity of 1 and (x-3) is a zero of multiplicity of 2.
(https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-intervals/a/zeros-of-polynomials-and-their-graphs)
f(x) = (x-1)(x-3)(x-3) is a third degree polynomial with 2 zeros, where (x-1) is a zero with multiplicity of 1 and (x-3) is a zero of multiplicity of 2.
For a root with an odd multiplicity, the function will have a sign change and cross the x-axis. For roots with an even multiplicity, the function will touch the x-axis but not change signs. In f(x) = (x-1)(x-3)(x-3), the graph touches the x axis and swoops back in the same, positive direction it came from.
(https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-intervals/v/polynomial-zero-multiplicity)
Examples
The root a of a polynomal p(x) has multiplicity m if (x - a)m is the largest power of x-a that is a factor of p. In other words, p(x) = (x-a)ms(x), where s(x) is a polynomial for which a is not a root.
If a is a root of multiplicity 5 for the polynomial p, and a root of multiplicity 7 for the polynomial q, what might limx-->ap(x) / q(x) be?
If a is a root of multiplicity 8 for the polynomial p, and a root of multiplicity 4 for the polynomial q, what might limx-->ap(x) / q(x) be?
- The numerator wins the race to 0, so the limit is 0.
- ANS: 0
If a is a root of multiplicity 6 for the polynomial p, and a root of multiplicity 6 for the polynomial q, what might limx-->ap(x) / q(x) be?