Question

The degree of the polynomial function f(x) is 4. The roots of the equation f(x)=0 are −2, −1, 1 and 3.

Answer 1

Answer:

f(x) = x^4 - x^3 - 7x^2 + x + 6

Step-by-step explanation:

Please, always share the instructions with every question you post.  I am assuming that you are to write out the factors corresponding to the given roots, thus constructing a 4th degree polynomial, and then performng the multiplication necesssary to produce that polynomial with terms in descending order by power of x.  Note that there must be 5 such terms.

f(x) = (x + 2)(x + 1)(x - 1)(x - 3)

Multiplying this out, we get:  

f(x) = (x + 2)(x^2 - 1)(x - 3), or

f(x) =      (x^3 - x + 2x^2 - 2)(x - 3), or:

f(x) = (x^3 + 2x^2 - x - 2)(x - 3), or:

f(x) = x^4 + 2x^3 - x^2 - 2x - 3x^3 - 6x^2 + 3x + 6

Combining like terms, we get:

f(x) = x^4 - x^3 - 7x^2 + x + 6

Note that because this is a 4th degree polynomial, the end behavior resembles that of the graph of y = x^4.  As x increases, y initially decreases in Quadrant II, passes through the y-intercept (0, 6), and then increases without bound in Quadrant I.