# Find all the zeros of f(x).  f(x) = x^4 - 4x^3 - 7x^2 + 28x First factor out an x, then get the p/q list. List answers from smallest to largest.  If there is a double root, list it twice....

Find all the zeros of f(x).  f(x) = x^4 - 4x^3 - 7x^2 + 28x

First factor out an x, then get the p/q list.

List answers from smallest to largest.  If there is a double root, list it twice.  Keep fractions in fractional form.

ishpiro | Certified Educator

Since `f(x) = x^4 -4x^3 -7x^2 + 28x` is a fourth degree polynomial, there can be up to 4 real zeros, but some of them might be the same.

As suggested, start by factoring out x:

`f(x) = x(x^3 -4x^2 -7x + 28)`

So the first zero is `x = 0` . Now we are looking for zeros of the cubic polynomial `x^3-4x^2-7x+28` .

"p/q" list refers to all possible numbers such the numerator, p, is a factor of the constant term (here, 28) and the denominator, q, is a factor of the leading coefficient (here, 1). Since the leading coefficient is 1 the only possible value for q is 1, so all zeros of this polynomial are integers.

All possible factors of 28 are `p=+- 1, +-2, +-4, +-7, +-28` .

Check which one of these numbers is a zero of the above polynomial.

1: `1^3 -4*1^2 -7*1 - 28 != 0` , so 1 is not a zero

-1: `(-1)^3 -4*(-1)^2 -7*(-1) - 28 != 0`

so -1 is not a zero

Similarly, `+-2` are not zeros either.

However, pluggin in x = 4 yields a zero. We can also notice that the polynomial can be factored by grouping:

`x^3-4x^2-7x+28=x^2(x-4)-7(x-4) = (x-4)(x^2 - 7)`

So x = 4 is a zero and two remaining zeros can be found by setting

`x^2 - 7 = 0`

This results in `x= +-sqrt(7)` .

Therefore, the zeros of f(x), listed the results in increasing order, are

`-sqrt(7), 0, sqrt(7), 4` .