Chapter 1.3, Problem 68PS

Solution

a.

To write: A quadratic polynomial that represents the area of the diagram.

A quadratic polynomial that represents the area of the diagram is $x^2+5x+6$ .

Given:

  

Calculation:

Consider the following figure.

  

A quadratic polynomial that represents the area of the diagram is $x^2+5x+6$ .

b.

To factor: The expression $x^2+5x+6$ and explain how the diagram models the factorization.

The expression $x^2+5x+6$ can be factorized as $x^2+5x+6=\left(x+2\right)\left(x+3\right)$ .

Given:

  

Calculation:

To find integers $m$ and $n$ to factor $x^2+bx+c$, such that:

  $\begin{array}{c}{x}^2+bx+c=\left(x+m\right)\left(x+n\right)\\ =x^2+\left(m+n\right)x+mn\end{array}$

So, the sum of $m$ and $n$ must equal $b$ and the product of $m$ and $n$ must equal $c$ .

For $x^2+5x+6=\left(x+m\right)\left(x+n\right)$ , $mn=6$ and $m+n=5$ .

$m,n$	$1,6$	$-1,-6$	$2,3$	$-2,-3$
$m+n$	$7$	$-7$	$5$	$-5$

From the table, the fourth column is suitable to factorization of $x^2+5x+6$ .

  $m=2,n=3$

Hence, $x^2+5x+6=\left(x+2\right)\left(x+3\right)$ .

c.

To draw: A diagram that models the factorization $x^2+8x+15=\left(x+5\right)\left(x+3\right)$ .

A diagram that models the factorization $x^2+8x+15=\left(x+5\right)\left(x+3\right)$ is as follows.

  

Given:

  

Diagram:

The diagram models the factorization $x^2+5x+6=\left(x+2\right)\left(x+3\right)$ is that the area that initially measures $x$ units by $x$ units wants to be adding $2$ units to the length and $3$ units to the width.

A diagram that models the factorization $x^2+8x+15=\left(x+5\right)\left(x+3\right)$ is as given below.