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VOOZH | about |
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VOOZH | about |
Number System is a method of representing numbers on a number line. The symbols range from 0-9 and are termed digits. A polynomial is a function of the form f(x) = an xn + anā1 xnā1 + ... + a2x2 + a1x + a0. The degree of a polynomial is the highest power of x in the expression. Constant (non-zero) polynomials are of degree 0, linear polynomials (maximum power of x is 1) are of degree 1, Quadratic polynomials (maximum power of x is 2) are of degree 2 and so on.
If a and b are roots, then the polynomial function with these roots is f(x) = (x - a)(x - b), or a multiple of this. For example, if a quadratic expression has the roots x = 3 and x = -2, then the function must be f(x) = (x - 3)(x + 2), or a constant multiple of this. This can be applied to polynomials of any degree. For example, if the roots of a polynomial are x = 2, x = 3, x = 4, then the function must be f(x) = (x - 2)(x - 3)(x - 4), or a constant multiple of this. Lets also try to think about the function f(x) = (x - 1)2 . It is seen that x - 1 = 0, so x = 1. For this function, there is a root. This is what is called a repeated root and this root can be repeated any number of times. For example, f(x) = (x - 2)3(x + 4)4 has a repeated root x = 2, and a repeated root x = -4. It can be said that the root x = 2 has a multiplicity of 3 and that the root x = -4 has a multiplicity of 4.
Multiple roots of a polynomial are roots whose factors show up more than once in the complete factorization of the polynomial. We call the number of times a factor shows up in the complete factorization the multiplicity of the root. The following examples will demonstrate how multiplicity and multiple roots are found.
Solution:
The Objective here to is to find the Multiple zero and multiplicity of f(x) = x3 + 2x2 + x. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. A multiple zero is a root with multiplicity m ā„ 2.
f(x) = x3 + 2x2 + x. Will be equated to zero.
x3 + 2x2 + x = 0
x(x2 + 2x + 1) = 0 (extract x common from the equation and the remaining part becomes a quadratic equation)
x2 + 2x + 1 can be written as (x + 1)2 it can be seen that the roots or zeroes of f(x) are 0, -1. Here zero has a multiplicity of 1 since it occurs once in the factored form. -1 has a multiplicity of 2. Therefore, multiple zero of f(x) = x3 + 2x2 + x, is -1 and it has multiplicity of 2.
Question 1: What is the multiple zero and multiplicity of y = 3(x + 3)3 (x + 2)4 (x - 1)2 (x - 5).
Solution:
Roots of this function are,
x + 3 = 0 -> x = -3
x + 2 = 0 -> x = -2
x - 1 = 0 -> x = 1
x - 5 = 0 -> x = 5
Multiple zeroes are -5, -2, 1. Multiplicity of x = -5 is 3 because x + 5 is raised to the power 3, Similarly, x = -2 is 4 and x = 1 is 2.
Question 2: What is the multiple zero and multiplicity of y = (x + 1)2(x + 3)3
Solution:
Roots of this function are,
x + 1 = 0 -> x = -1
x + 3 = 0 -> x = -3F
Multiple zeroes are -1, 3. The multiplicity of x = -1 is 2 because x + 1 term is raised to the power 2 and multiplicity of x = -3 is 3 because x + 3 is raised to the power 3
