Respuesta :
The polynomial of 3rd degree whose zeros and degree are given as for zero of (-2), multiplicity 1 and for zero of (3), multiplicity 2 is [tex]f(x)=(x^{3}-10x^{2} -15x+18)[/tex]
As per the question statement, a 3rd degree polynomial has a zero of (-2), multiplicity 1 and another zero of (3), multiplicity 2.
We are required to determine the polynomial.
Since, (-2) is a zero of the polynomial, then [tex][x-(-2)]=(x+2)[/tex] will be a factor of the polynomial. And since for zero of (-2), multiplicity is 1, this means that the factor of (x + 2) is only multiplied once.
Similarly, as (3) is another zero of the polynomial, then [tex](x-3)[/tex] will be a factor of the polynomial. And since for zero of (3), multiplicity is 2, this means that the factor of (x - 3) will be multiplied twice, i.e., [tex](x-3)^{2}[/tex].
Therefore, our function, say, f(x) will be [tex](x+2)(x-3)^2[/tex]
[tex]or, f(x) = (x+2)(x-3)^2\\or, f(x) = (x + 2)(x^{2} +9-12x)\\or,f(x) =x^{3}+9x-12x^{2} +2x^{2} +18-24x\\ or, f(x)=x^{3}+(2x^{2}-12x^{2})+(9x-24x)+18\\or, f(x)=(x^{3}-10x^{2} -15x+18)[/tex]
Hence, our required polynomial is [tex]f(x)=(x^{3}-10x^{2} -15x+18)[/tex]
- Polynomial: A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s)
- Degree of a Polynomial: The highest value of power of the variable of a polynomial is known as the degree of the polynomial.
- Zero of a Polynomial: The value of the variable for which, the polynomial or equation equates to zero, is known as Zero of the Polynomial.
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