Question

Select all zeros of the polynomial function.

g(x)=x4 + 4x³ + 7x² + 16x + 12
-4
1
-5i
i
-3
2
-4i
21
-2
3
-3i
3i
-1
4
-2i
4i
0
5
-i
5i

Answer 1

Hi,

`Zeros\ are\ -1,-3,2i,-2i\\`

Explanation:

To find the zeros of a polynomial function, we need to solve the equation g(x) = 0.

The given polynomial function is g(x) = x^4 + 4x^3 + 7x^2 + 16x + 12.

To solve this equation, we can use different methods such as factoring, synthetic division, or the rational root theorem. In this case, factoring or synthetic division might not be easy to apply, so we will use the rational root theorem to find the possible rational roots.

The rational root theorem states that if a polynomial has a rational root p/q, where p is a factor of the constant term (12) and q is a factor of the leading coefficient (1), then p/q is a possible root of the polynomial.

The factors of 12 are ±1, ±2, ±3, ±4, ±6, and ±12.

The factors of 1 are ±1.

Using the rational root theorem, the possible rational roots are:

±1, ±2, ±3, ±4, ±6, and ±12.

Now, we can check each possible root by substituting it into the polynomial function g(x) and see if it equals zero.

By trying out the possible roots, we find that g(−1) = 0. This means that -1 is a zero of the polynomial function.

So, the zeros of the polynomial function g(x) = x^4 + 4x^3 + 7x^2 + 16x + 12 are x = -1.

Please note that this method only gives us the rational zeros. There might be additional irrational or complex zeros that we haven't found using this method.

`g(x)=x^4+4x^3+7x^2+16x+12\\=x^4+x^3+3x^3+3x^2+4x^2+4x+12x+12\\=x^3(x+1)+3x^2(x+1)+4x(x+1)+12(x+1)\\=(x+1)(x^3+3x^2+4x+12)\\=(x+1)(x^2(x+3)+4(x+3))\\=(x+1)(x+3)(x^2+4)\\\\=(x+1)(x+3)(x-2i)(x+2i)\\\\Zeros\ are\ -1,-3,2i,-2i\\`