Chapter 6.7, Problem 48PPS

To determine

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

Answer to Problem 48PPS

The equation of given zeroes $-2,-3,4-3i$ is $\boxed{f\left(x\right)=x^4-3x^3-9x^2+77x+150}$ .

Explanation of Solution

Given information:

The given zeros are $-2,-3,4-3i$ .

Remember, if $4-3i$ is a zero, then $4+3i$ is also a zero according to conjugates theorem. Therefore, the factors of the polynomial functions are:

  $f\left(x\right)=\left(x-\left(4+3i\right)\right)\left(x-\left(4-3i\right)\right)\left(x+2\right)\left(x+3\right)$

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

  $f\left(x\right)=\left(\left(x-4\right)-3i\right)\left(\left(x-4\right)+3i\right)$ Regroup the terms

  $f\left(x\right)=\left({\left(x-4\right)}^2-9i^2\right)$ Use difference of two squares.

  $f\left(x\right)=x^2-8x+16-9\left(-1\right)$ Square the terms

  $f\left(x\right)=x^2-8x+25$ Simplify

  $f\left(x\right)=\left(x^2-8x+25\right)\left(x+2\right)$ Multiply the 3^rd binomial with the result.

  $f\left(x\right)=\left(x\right)\left(x^2-8x+25\right)+\left(2\right)\left(x^2-8x+25\right)$ Use Distributive property

  $f\left(x\right)=x^3-8x^2+25x+2x^2-16x+50$ Multiply

  $f\left(x\right)=x^3-8x^2+2x^2+25x-16x+50$ Rearrange terms.

  $f\left(x\right)=x^3-6x^2+9x+50$ Combine like terms.

  $f\left(x\right)=\left(x^3-6x^2+9x+50\right)\left(x+3\right)$ Multiply result with 4^th term

  $f\left(x\right)=\left(x\right)\left(x^3-6x^2+9x+50\right)+\left(3\right)\left(x^3-6x^2+9x+50\right)$ Use Distributive property

  $f\left(x\right)=x^4-6x^3+9x^2+50x+3x^3-18x^2+27x+150$ Multiply

  $f\left(x\right)=x^4-6x^3+3x^3+9x^2-18x^2+50x+27x+150$ Rearrange terms

  $\boxed{f\left(x\right)=x^4-3x^3-9x^2+77x+150}$ Simplify