Find a polynomial function of degree 7 with -2 as a zero of multiplicity 3, 0 as a zero of multiplicity 3, and 2 as a zero of multiplicity 1.
f(x)=
Gracias
To find a polynomial function with the given zeros and multiplicities, we can start by writing the factors of the polynomial. Since -2 is a zero of multiplicity 3, 0 is a zero of multiplicity 3, and 2 is a zero of multiplicity 1, we can write the factors as follows:
(x + 2)^3 * x^3 * (x - 2)
To find the polynomial function, we multiply these factors together:
f(x) = (x + 2)^3 * x^3 * (x - 2)
Expanding this expression gives:
f(x) = (x^3 + 6x^2 + 12x + 8) * x^3 * (x - 2)
Simplifying further:
f(x) = (x^6 + 6x^5 + 12x^4 + 8x^3) * (x - 2)
Multiplying again:
f(x) = x^7 - 2x^6 + 6x^6 - 12x^5 + 12x^5 - 24x^4 + 8x^4 - 16x^3
Combining like terms:
f(x) = x^7 - 2x^6 + 4x^6 - 16x^4 - 8x^3
Thus, the polynomial function of degree 7 with the given zeros and multiplicities is:
f(x) = x^7 - 2x^6 + 4x^6 - 16x^4 - 8x^3
Gracias!
To find a polynomial function with the given zeros and multiplicities, we can start by writing the factors of the polynomial based on the zeros.
From the given information, we know that -2 is a zero with multiplicity 3, 0 is a zero with multiplicity 3, and 2 is a zero with multiplicity 1.
Since -2 is a zero with multiplicity 3, we can write the factor (x + 2)^3.
Since 0 is a zero with multiplicity 3, we can write the factor (x - 0)^3, which simplifies to x^3.
Since 2 is a zero with multiplicity 1, we can write the factor (x - 2)^1, which simplifies to (x - 2).
Now we can multiply these factors together to find the polynomial function:
f(x) = (x + 2)^3 * x^3 * (x - 2)
Expanding this expression, we get:
f(x) = (x + 2)(x + 2)(x + 2) * x^3 * (x - 2)
= (x + 2)^3 * x^3 * (x - 2)
Therefore, the polynomial function of degree 7 with -2 as a zero of multiplicity 3, 0 as a zero of multiplicity 3, and 2 as a zero of multiplicity 1 is:
f(x) = (x + 2)^3 * x^3 * (x - 2)
Gracias!