Chapter 6.7, Problem 43PPS

To determine

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

Answer to Problem 43PPS

The equation of given zeroes $5,-2,-1$ is $\boxed{f\left(x\right)=x^3-2x^2-13x-10}$ .

Explanation of Solution

Given information:

The given zeros are $5,-2,-1$ .

Remember that the given zeros are also the roots of the function. Therefore, factors of the polynomial function s are:

  $f\left(x\right)=\left(x-5\right)\left(x+2\right)\left(x+1\right)$

To know polynomial function, multiply first two binomials using FOIL method, then multiply the third binomial using horizontal method.

  $f\left(x\right)=x\left(x\right)+x\left(2\right)+\left(x\right)\left(-5\right)+\left(-5\right)\left(2\right)$ Use the FOIL method

  $f\left(x\right)=x^2+2x-5x-10$ Multiply

  $f\left(x\right)=x^2-3x-10$ Combine like terms

  $f\left(x\right)=\left(x^2-3x-10\right)\left(x+1\right)$ Multiply the 3^rd binomial with the result.

  $f\left(x\right)=x\left(x^2-3x-10\right)+1\left(x^2-3x-10\right)$ Use distributive property.

  $f\left(x\right)=x^3-3x^2-10x+x^2-3x-10$ Multiply

  $f\left(x\right)=x^3-3x^2+x^2-10x-3x-10$ Rearrange terms.

  $\boxed{f\left(x\right)=x^3-2x^2-13x-10}$ Combine like terms