Chapter 8.5, Problem 45PPE

To determine

the standards form of the given polynomial model and also find the factors for the standard expression

Answer to Problem 45PPE

The standard form of the given polynomial model is $20x^2+60x+25$

The factors of the standard form are $5.\left(2x+1\right).\left(2x+5\right)$

Explanation of Solution

Given info:

The polynomial model form is given as follows:

  

Formula used:

Show the expression of standard form as follows:

  $a{x}^2+bx+c$

Here, $a,b,c$ are integers, $b$ is the value which depends on the multiplies $\left(x,y\right)$ of the product of $a$ and $c$ .

Condition of $b$ value depends on $c$ value. If the value $c$ is negative, then $b=x-y$ . If the value $c$ is positive, then $b=x+y$ .

Calculation:

1. Find the polynomial from the given model as follows:

Add the values in the rectangular model at right and bottom side which gives the polynomial terms.

• Right side of the rectangular model - $10x+5$
• Bottom side of the rectangular model - $2x+5$
2. Find the standard form by multiplying the polynomial terms as follows:

  $\begin{array}{l}\left(10x+5\right)\times \left(2x+5\right)\\ \left(20x^2+50x+10x+25\right)\\ \left(20x^2+60x+25\right)\end{array}$

3. Find the factors of the standard form as follows:

  $\begin{array}{l}\left(20x^2+60x+25\right)\\ 5\left(4x^2+12x+5\right)\\ 5\left(4x^2+10x+2x+5\right)\\ 5\left(2x\left(2x+5\right)+1\left(2x+5\right)\right)\\ 5.\left(2x+1\right).\left(2x+5\right)\end{array}$

Hence, the standard form is $\left(20x^2+60x+25\right)$ and the factors of the standard form are $5.\left(2x+1\right).\left(2x+5\right)$

Conclusion:

Thus, the standard form is $\left(20x^2+60x+25\right)$ and the factors of the standard form are $5.\left(2x+1\right).\left(2x+5\right)$