Respuesta :
Given:
Zero: 2, multiplicity: 3
Zero: 0, multiplicity: 2
Degree: 5
Leading coefficient = 1
To find:
The polynomial function.
Solution:
The general form of a polynomial is
[tex]P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}[/tex]
where, a is a constant, [tex]c_1,c_2,...,c_n[/tex] are zeroes with multiplicity [tex]m_1,m_2,...,m_n[/tex].
Using the given information and the general form of a polynomial, we get
[tex]P(x)=a(x-2)^{3}(x-0)^{2}[/tex]
[tex]P(x)=ax^2(x^3-6x^2+12x-8)[/tex]
[tex]P(x)=a(x^5-6x^4+12x^3-8x^2)[/tex]
Leading coefficient is 1, so the value of a is also 1.
[tex]P(x)=1(x^5-6x^4+12x^3-8x^2)[/tex]
[tex]P(x)=x^5-6x^4+12x^3-8x^2[/tex]
Therefore, the required polynomial is [tex]P(x)=x^5-6x^4+12x^3-8x^2[/tex].
The polynomial for the conditions with leading coefficient 1 that has the given zeros, multiplicities, and degree. Zero: 2, multiplicity: 3 Zero: 0, multiplicity: 2 Degree: 5 is given as
[tex]\rm\bold{ y(x) = 1\times (x-2)^3(x-0)^2 }[/tex]
According to the given condition
The given conditions to write the polynomial equations are as follows
Zero 2 and Multiplicity 3
Zero 0 and Multiplicity 2
Degree of polynomial expression = 5
Leading coefficient of polynomial expression = 1
Let us consider the polynomial equation has only one variable and hence in general form we can write the equation of polynomial as follows
[tex]\rm y (x) = a(x-\alpha)^m (x-\beta)^n........(1)\\Where \\a = Leading \; coefficient\; of\; polynomial\; expression \\\alpha ,\beta = Roots \; of \; the\; polynomial\; expression\\m\; and \; n \; are \; multiplicities \; of \; the \; roots \; \alpha,\beta \; respectively\\Degree = m+n[/tex]
From the general form of polynomial expression as shown in equation (1) we can write
Polynomial expression for the given conditions is formulated in equation (2)
[tex]\rm y(x) = 1\times (x-2)^3(x-0)^2 .......(2)[/tex]
So the polynomial for the conditions given in question is expressed as
[tex]\rm\bold{ y(x) = 1\times (x-2)^3(x-0)^2 }[/tex]
For more information please refer to the link given below
https://brainly.com/question/15301188
