Find a polynomial function of degree 4 with-2 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1.
f(x)=
Gracias
(x+2)^3 * x
To find a polynomial function with the given zeros and multiplicities, you can start by utilizing the concept of zero product property. The zero product property states that if a polynomial function has a zero at a particular value, then the polynomial can be factored by the binomial (x - a), where 'a' is the zero.
In this case, we have a zero of -2 with a multiplicity of 3, which means (x + 2) will be a factor of the polynomial raised to the power of 3: (x + 2)^3. We also have a zero of 0 with a multiplicity of 1, which means (x - 0) = x is a factor of degree 1.
Multiplying these factors together, we get:
f(x) = (x + 2)^3 * x
To simplify it further, we can expand the cube of (x + 2) using the binomial theorem:
f(x) = (x + 2)(x + 2)(x + 2) * x
= (x^2 + 4x + 4)(x + 2) * x
= (x^3 + 4x^2 + 4x + 2x^2 + 8x + 8) * x
= x^4 + 6x^3 + 12x^2 + 8x
So, the polynomial function of degree 4 with -2 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1 is:
f(x) = x^4 + 6x^3 + 12x^2 + 8x
Espero que esto te ayude.