What is a polynomial function in standard form with zeros 1, 2, -2, and -3?
To find the polynomial function in standard form with the given zeros, we can start by using the zero-product property.
Since the zeros are 1, 2, -2, and -3, we know that the factors of the polynomial are (x - 1), (x - 2), (x + 2), and (x + 3).
To find the polynomial function, we can multiply these factors:
(x - 1)(x - 2)(x + 2)(x + 3)
Expanding this expression, we get:
(x^2 - 3x + 2)(x^2 + 2x - 6)
Multiplying further, we have:
(x^4 - 3x^3 + 2x^2 + 2x^3 - 6x^2 + 4x)(x^2 + 2x - 6)
Combining like terms, we get:
(x^4 - x^3 - 4x^2 + 4x)(x^2 + 2x - 6)
Expanding again, we have:
x^6 + 2x^5 - 6x^4 - x^5 - 2x^4 + 6x^3 - 4x^4 - 8x^3 + 24x^2 + 4x^3 + 8x^2 - 24x
Combining like terms one final time, we get:
x^6 + x^5 - 12x^4 + 9x^3 + 24x^2 - 24x
Therefore, the polynomial function in standard form with zeros 1, 2, -2, and -3 is:
f(x) = x^6 + x^5 - 12x^4 + 9x^3 + 24x^2 - 24x