Chapter 8.6, Problem 1MP

(a)

To determine

To find: A trinomial that represents the total area of the two plots

Answer to Problem 1MP

  $2x^2+17x+30$ is the total area of two plot

Explanation of Solution

Given Information:

The area of Kelly’s plot: $x^2+8x+12$

The area of Roberto’s plot: $x^2+9x+18$

Formula Used:

Total area of two plots is the sum of the area of Kelly’s plot and area of Roberto’s plot.

Calculation:

Total area of two plots =area of Kelly’s plot + area of Roberto’s plot

  $=\left(x^2+8x+12\right)+\left(x^2+9x+18\right)$

  $=\left(x^2+x^2\right)+\left(8x+9x\right)+\left(12+18\right)$

  $=2x^2+17x+30$

(b)

To determine

To factor: the trinomial found in part (a)

Answer to Problem 1MP

  $2x^2+17x+30$

  $=\left(x+6\right)\left(2x+5\right)$

Explanation of Solution

Given Information:

The polynomial that represents the total area of the two plots is $2x^2+17x+30$

Formula Used:

• A trinomial $a{x}^2+bx+c$ can be expressed as a product of two binomials
• Distributive property
$\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

Calculation:

Consider the trinomial $2x^2+17x+30$ and write $17x\text{ as }\left(12x+5x\right)$

  $2x^2+17x+30$

  $=2x^2+12x+5x+30$

  $=\left(2x^2+12x\right)+\left(5x+30\right)$

  $=2x\left(x+6\right)+5\left(x+6\right)$ [Taking 2x from first bracket and 5 from second pair]

  $=\left(x+6\right)\left(2x+5\right)$ [Distributive property]

∴$2x^2+17x+30$ is expressed as product of two binomials $\left(x+6\right)$ and $\left(2x+5\right)$

(c)

To determine

To interpret: the result in part (b)

Explanation of Solution

Given Information:

  $2x^2+17x+30$ is expressed as product of two binomials $\left(x+6\right)$ and $\left(2x+5\right)$

  $2x^2+17x+30$ = $\left(2x+5\right)$

  $\left(x+6\right)$

From part (a) $2x^2+17x+30$ is the total area of two plots.

Since, area is the product of length and breadth

  $\left(2x+5\right)$ and $\left(x+6\right)$ represents length and breadth of combined plot

Kelly and Roberto has decided to combine their respective plot to make a rectangular plot.

They plan to place the two x ft by x ft square flower bed side by side to make a 2xft by x ft flower bed, centered inside a rectangular border.

Length of the new plot = $\left(2x+5\right)$

Width of the new plot = $\left(x+6\right)$