Question

Use the given zero to find the remaining zeros of the function.h(x)=x^4−18x^3+84x^2+168x−2080   zero: 6−4i

Answer 1

The remaining zeros are: -4, 6+4i, 10

Step-by-step explanation

By the fundamental theorem of algebra, an n-degree polynomial with complex coefficients has exactly n zeros. This polynomial function is a 4th-degree polynomial, so it will have 4 zeros.

Also, if a complex number is a zero, its complex conjugate is a zero of the polynomial. This means that x = 6 + 4i is also a zero.

Remember that a polynomial with zeros x1, x2, x3, x4 can be written as the product of its factors,

`p(x)=(x-x_1)(x-x_2)(x-x_3)(x-x_4)_{}`

For now, we have two of the zeros, so we have two factors: (x - 6 + 4i) and (x - 6 - 4i). The function can be divided by the product of these factors,

`(x-6+4i)(x-6-4i)=\lbrack(x-6)+(4i)\rbrack\lbrack(x-6)-(4i)\rbrack`

Note that this has the form of a difference of two squares,

`\lbrack(x-6)+(4i)\rbrack\lbrack(x-6)-(4i)\rbrack=(x-6)^2-(4i)^2`

Expand the binomial and solve the squares,

`\begin{gathered} (x-6)^2-(4i)^2=(x^2-2\cdot6\cdot x+6^2)-4^2\cdot i^2 \\ \\ (x-6)^2-(4i)^2=x^2-12x+36-(-1)\cdot16 \\ \\ (x-6)^2-(4i)^2=x^2-12x+36+16 \\ \\ (x-6)^2-(4i)^2=x^2-12x+52 \end{gathered}`

Let's use long division to divide the function h(x) by this factor,

This result means that we can write the polynomial as,

`h(x)=(x^2-6x-40)(x^2-12x+52)`

The second factor is the one we got before after multiplying the complex factors, and the first factor is the one we will use to find the other two zeros. We have to solve,

`x^2-6x-40=0`

For which we can use the quadratic formula,

`\begin{gathered} ax^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \end{gathered}`

In this case, a = 1, b = -6, and c = -40,

`x=\frac{-(-6)\pm\sqrt[]{(-6)^2-4\cdot1\cdot(-40)}}{2\cdot1}=\frac{6\pm\sqrt[]{36+160}}{2}=\frac{6\pm\sqrt[]{196}}{2}=(6\pm14)/(2)`

We have the other two zeros, which are real,

`\begin{gathered} x=(6+14)/(2)=(20)/(2)=10 \\ \\ x=(6-14)/(2)=(-8)/(2)=-4 \end{gathered}`

In summary, the remaining zeros of the function are -4, 6+4i, 10.