Chapter 6.8, Problem 3SGR

To determine

Whether the statement “Polynomials that has a degree one less than the original polynomial is called $\text{prime polynomial}$ .” is true or false.

Answer to Problem 3SGR

The statement “Polynomials that has a degree one less than the original polynomial is called $\text{prime polynomial}$ .” is false.

Explanation of Solution

Given information:

The statement “Polynomials that has a degree one less than the original polynomial is called $\text{prime polynomial}$ .”

Consider the provided statement “Polynomials that has a degree one less than the original polynomial is called $\text{prime polynomial}$ .”

The polynomials that cannot be factored are called $\text{Prime polynomials}$ .

The $\text{depressed polynomial}$ has a degree that is one less than the original polynomial.

A polynomial is in the form of: $4x{y}^2+3x-5$ .

The polynomials that results of diving something from an original polynomial.

For example: One root of $x^4+5x^3+3x^2-8x+3=0$ is $-3$ . Find the depressed equation.

Divide $x^4+5x^3+3x^2-8x+3=0$ by $x+3$ , by using long division.

So, the expression can be simplified as,

  $\begin{array}{l}\left(x^4+5x^3+3x^2-8x+3\right)÷\left(x+3\right)\\ x^4+5x^3+3x^2-8x+3=0\end{array}$

Observe to solve the expression first open the brackets.

  $x+3\begin{array}{c}\hfill x^3+2x^2-3x+1\\ \hfill \overline{)\begin{array}{l}\begin{array}{ccccccccc}{x}^4& +& 5x^3& +& 3x^2& -& 8x& +& 3\end{array}\\ \begin{array}{ccc}{x}^4& +& 3x^3\end{array}\\ \begin{array}{ccc}2x^3& +& 3x^2\\ 2x^3& +& 6x^2\end{array}\\ \begin{array}{cccc}-& 3x^2& -& 8x\\ & 3x^2& +& 9x\end{array}\\ \begin{array}{cccc}& x& +& 3\\ -& x& -& 3\end{array}\\ \begin{array}{c}0\end{array}\end{array}}\end{array}$

Now since, $\text{Dividend=Quotent×Divisor+Remainder}$ . Here, dividend is $x^4+5x^3+3x^2-8x+3$ and quotient is $x^3+2x^2-3x+1$ , divisor is $x+3$ .

Therefore, $x^4+5x^3+3x^2-8x+3=\left(x+3\right)\left(x^3+2x^2-3x+1\right)+0$ .

Then the depressed equation is: $x^3+2x^2-3x+1$

The example of Prime polynomial is:

  $x^2+3x+2$

This equation is not a prime polynomial as it can be written in the form of, $\left(x+1\right)\left(x+2\right)$ .

Thus, the statement “Polynomials that has a degree one less than the original polynomial is called $\text{prime polynomial}$ .” is false.