Chapter 4.5, Problem 1E

To determine

To complete: the sentence.

Answer to Problem 1E

If a polynomial function $f$ has integer coefficients, then every rational solution of $f\left(x\right)=0$ has the form $\frac{p}{q}$ , where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$ .

Explanation of Solution

Given information:

Given sentence:

If a polynomial function $f$ has integer coefficients, then every rational solution of $f\left(x\right)=0$ has the form $\frac{p}{q}$ , where $p$ is a factor of the ----------and $q$ is a factor of the -----------.

Calculation:

The rational root theorem states that “if $f\left(x\right)=a_n{x}^n+\mathrm{...}+a_{1}x+a_0$ has integer coefficient, then every rational solution of $f\left(x\right)=0$ has the form.

  $\frac{p}{q}=\frac{\text{Factor}\text{of}\text{constant}\text{term}{a}_0}{\text{Fcator}\text{of}\text{leading}\text{coefficient}{a}_n}$

Therefore, fill the blank spaces as “If a polynomial function $f$ has integer coefficients, then every rational solution of $f\left(x\right)=0$ has the form $\frac{p}{q}$ , where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$ ”.