Question

Write the simplest polynomial function with integral coefficients that has the give. Zeros

Answer 1

Step 1:

Write the zeros of the polynomial

$\text{x = 7, x = }\pm\text{7i}$

Note: A complex zero must have both negative and positive values.

Step 2:

Write the factors of the polynomial.

$\begin{gathered} x\text{ = 7 and x = }\pm7i \\ x-7=0\text{ } \\ x^2\text{ = -49 is the samd as (x = }\pm7i) \\ x^2\text{ + 49 = 0} \\ \text{The thre}e\text{ factors are: x - 7, x - 7i , and x + 7i} \end{gathered}$

Step 3:

$\begin{gathered} \text{Multiply x - 7 and x}^2\text{ + 49 to get the simplest polynomial} \\ (x-7)(x^2\text{ + 49)} \\ x^3+49x-7x^2\text{ - 343} \\ x^3-7x^{2^{}}\text{ + 49x - 343} \end{gathered}$

Final answer

$x^3-7x^2\text{ + 49x - 343}$