Chapter 7.6, Problem 42E

To determine

To calculate: Factors of given polynomial modeled by the tiles using algebra tiles.

Answer to Problem 42E

a) $2x^2+5x-3=(2x-1)(x+3)$

b) $3x^2-2x-1=(3x+1)(x-1)$

Explanation of Solution

Given information: Tile structures of polynomials are given.

Calculation:

By give tile structure, it is clear that the given polynomial is $2x^2+5x-3$ .

Now, $2x^2+5x-3=2x^2-x+6x-3$

So, using algebra tiles, the given polynomial can be modeled as below:

  $\begin{array}{llll}\hfill & \begin{array}{ll}+\hfill & \hfill \end{array}\hfill & \begin{array}{ll}+\hfill & \hfill \end{array}\hfill & \begin{array}{l}-\hfill \end{array}\hfill \\ \begin{array}{l}+\hfill \\ \hfill \end{array}\hfill & \begin{array}{ll}+\hfill & \hfill \\ \hfill & \hfill \end{array}\hfill & \begin{array}{ll}+\hfill & \hfill \\ \hfill & \hfill \end{array}\hfill & \begin{array}{c}-\\ \end{array}\hfill \\ \begin{array}{l}+\hfill \end{array}\hfill & \begin{array}{cc}+& \end{array}\hfill & \begin{array}{cc}+& \end{array}\hfill & \begin{array}{c}-\end{array}\hfill \\ \begin{array}{l}+\hfill \end{array}\hfill & \begin{array}{cc}+& \end{array}\hfill & \begin{array}{cc}+& \end{array}\hfill & \begin{array}{c}-\end{array}\hfill \\ \begin{array}{l}+\hfill \end{array}\hfill & \begin{array}{cc}+& \end{array}\hfill & \begin{array}{cc}+& \end{array}\hfill & \begin{array}{c}-\end{array}\hfill \end{array}$

Writing the polynomial in factored form using the dimensions of the rectangle,

  $2x^2+5x-3=(2x-1)(x+3)$

b) By give tile structure, it is clear that the given polynomial is $3x^2-2x-1$ .

Now, $3x^2-2x-1=3x^2-3x+x-1$

So, using algebra tiles, the given polynomial can be modeled as below:

  $\begin{array}{ccc}& \begin{array}{cc}+& \end{array}& \begin{array}{c}-\end{array}\\ \begin{array}{c}+\\ \end{array}& \begin{array}{cc}+& \\ & \end{array}& \begin{array}{c}-\\ \end{array}\\ \begin{array}{c}+\\ \end{array}& \begin{array}{cc}+& \\ & \end{array}& \begin{array}{c}-\\ \end{array}\\ \begin{array}{c}+\\ \end{array}& \begin{array}{cc}+& \\ & \end{array}& \begin{array}{c}-\\ \end{array}\\ \begin{array}{c}+\end{array}& \begin{array}{cc}+& \end{array}& \begin{array}{c}-\end{array}\end{array}$

Writing the polynomial in factored form using the dimensions of the rectangle,

  $3x^2-2x-1=(3x+1)(x-1)$