Find an equation of a polynomial function of degree 5 with integer coefficients with zeros 0, -2, and 1/2.
To find the equation of a polynomial function with integer coefficients and given zeros, we will use the concept of factoring. Since you're given the zeros 0, -2, and 1/2, we can write the factors of the polynomial as follows:
(x - 0) = x
(x - (-2)) = x + 2
(x - 1/2)
To get the equation of the polynomial, we multiply these factors together:
f(x) = (x)(x + 2)(x - 1/2)
Now, let's simplify and express the function in a more standard form:
f(x) = (x)(x + 2)(2x - 1) [multiplying through by 2 to eliminate the fraction]
Expanding this expression, we get:
f(x) = (2x² + 4x)(2x - 1)
Now, let's apply the distributive property and simplify further:
f(x) = (4x³ + 2x²) - (2x² + x)
Combining like terms, we get:
f(x) = 4x³ + (2x² - 2x²) + (2x - x)
Finally, the equation of the polynomial function of degree 5 with integer coefficients and zeros 0, -2, and 1/2 is:
f(x) = 4x³ + x
Note that the term with x is degree 1, not degree 5, because the remaining two zeros (0 and -2) each have multiplicity 1, so the total degree of the polynomial is 3, not 5.