Question

One of the zeros of the polynomial function is 3.

f(x)=x^4−x^3−7x^2+x+6

What is the factored form of the function?
a.) f(x)=(x−3)(x+1)(x+2)^2
b.) f(x)=(x−3)^2(x+1)(x+2)
c.) f(x)=(x−3)(x+1)(x−1)(x+2)
d.) f(x)=(x−3)(x+1)^2(x+2)

Answer 1

To find the factored form of the polynomial function f(x) = x^4 - x^3 - 7x^2 + x + 6, we need to determine the factors corresponding to the given zero of 3.

If 3 is a zero of the polynomial, it means that (x - 3) is a factor of the polynomial.

Using synthetic division or long division, we can divide the polynomial by (x - 3) to find the remaining factors:

3 | 1 -1 -7 1 6

| 3 6 -3 -6

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1 2 -1 -2 0

The result of the division is 1x^3 + 2x^2 - x - 2.

Now, we can further factor the remaining polynomial:

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

The quadratic factor x^2 + x - 2 can be factored as (x - 1)(x + 2).

Therefore, the factored form of the polynomial function f(x) = x^4 - x^3 - 7x^2 + x + 6 is:

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

So, the correct answer is c.) f(x) = (x - 3)(x + 1)(x - 1)(x + 2).