University:
University of North Carolina at Charlotte
Course:
MATH 1242 | Calculus II
Academic year:

2022
Views:

301

Pages:

2
Author:

Rosalinda G

MultiplicitiesSince f(x) = x+1 is a polynomial function of degree 1. you know that it has only one zero. It is trivial to see that the zero is -1.Now consider f(x) =(x+1)2 Since it is a polynomial function of degree 2. it must have two zeroes, and yet setting (x+1)2 equal to 0 and solving for xresults in only one zero: -1. How can this be?The Fundamental Theorem of Algebra guarantees that a polynomial function of degree n will have n zeroes, but they need not be all unique Some zeroesmay have multiplicity. In the case of f(x) = (x+1)2. the zero -1 has multiplicity of 2 since the factor x +1 is raised to the second powerIt is possible to look at the graph of a polynomial function and see if a zero has even or odd multiplicity, if in fact it has any. Consider the followingfunctions and their graphs. Each function has a zero at 3.MultiplicityFunction and GraphMultiplicityFunction and Graphof Zero 3of Zero 3yyAG6554433221120011.10123-3-2-1-1123456789-2-2-3-3-44-5-5-6-6y=(x-3)2y=x-3yy64A655443322110403-3-2-11913456789-3-2-1.1-2-2-3-3.4-4-5-5-6-6y=(x-3)y=(x-3)365 Function and GraphMultiplicityMultiplicityof Zero 3Function and Graphof Zero 3y4y66554433221100-21101-3-2 11123456789-2-2-3-3-4-4-5-5-6-6y=(x-3)5y=(x-3)6You can see that the function crosses the x-axis for zeroes with an odd number of multiplicities and remains on one side of the x-axis on either side ofazero with an even number of multiplicitiesExampleSuppose you were given the function f(x)=x*-2-4x2-4x+16 and told that one of its zeroes is 3,45, 2 2 Find the other zeroes of f(x).The ConjugatesZero Theorem tells you that 3_V5, is also a zero. Now use your graphing calculator to look at the graph of f(x) and see if it has any2 2real zerocs.yA765432103.2.1.012345-1-2-3The function touches the r-axis exactly once at x 2. so 2 is a zero of the function Based on the facts that there are no other real zeroes, that the functionhas four zeroes, that complex zeroes always occur in conjugate pairs, and that the function touched the x-axis at x = 2 but does not cross it, you canconclude that the zero 2 has a multiplicity of 2.Now you try.Find all of the zeroes of f(x)=x4+62+13x2+24x+36Select the link to check your answer.8Answer

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