Chapter 5.7, Problem 45PPS

To determine

Write a polynomial function of least degree with integral coefficients that have the given zeros.

Answer to Problem 45PPS

The equation of given zeroes $-1,-1,2i$ is $\boxed{f\left(x\right)=x^4+2x^3+5x^2+8x+4}$ .

Explanation of Solution

Given information:

The given zeros are $-1,-1,2i$ .

Remember, if $2i$ is a zero, then $-2i$ is also a zero according to conjugates theorem. Therefore, the factors of the polynomial functions are:

  $f\left(x\right)=\left(x-2i\right)\left(x+2i\right)\left(x+1\right)\left(x+1\right)$

To know polynomial function, multiply first two binomials having imaginary units. Then, multiply result with the third binomial, then with the fourth.

  $f\left(x\right)=x^2-4i^2$ Use difference of two squares.

  $f\left(x\right)=x^2-4\left(-1\right)$ Evaluate the imaginary unit, $i^2=-1$ .

  $f\left(x\right)=x^2+4$ Simplify.

  $f\left(x\right)=\left(x^2+4\right)\left(x+1\right)$ Multiply the 3^rd binomial with the result.

  $f\left(x\right)=x^2\left(x\right)+x^2\left(1\right)+4\left(x\right)+4\left(1\right)$ Use Foil Method

  $f\left(x\right)=x^3+x^2+4x+x$ Multiply.

  $f\left(x\right)=x\left(x^3+x^2+4x+4\right)+1\left(x^3+x^2+4x+4\right)$ Multiply the 4^th binomial with the result.

  $f\left(x\right)=x^4+x^3+4x^2+4x+x^3+x^2+4x+4$ Use distributive property

  $f\left(x\right)=x^4+x^3+x^3+4x^2+x^2+4x+4x+4$ Rearrange terms.

  $\boxed{f\left(x\right)=x^4+2x^3+5x^2+8x+4}$ Combine like terms.