Question

find each difference.

1. $(x^{2}+8x + 15)-(x^{2}-6x - 9)$
2. $(13mn^{2}-2nm^{2}+5n^{2})-(11mn^{2}-2n^{2})$
3. $(9y^{5}-6y^{3})-(6y^{5}-3y^{4}+5y^{3})$
4. $(10x^{2}-16x + 13)-(10x^{2}-15x - 10)$
5. the cost for a company to make a cell phone is $10x + 120$. the revenue that the company gets is $15x+75$. what is the total profit the company receives for selling $x$ cell phones?
6. if the same company in problem 5 sold 200 cell phones, what would their profit be?

optional reading: textbook lesson 16.1, check understanding & on your own 4, 6, 13 - 14; textbook lesson 16.2, check understanding & on your own 2, 9, 10, 12

Explanation:

Step1: Expand the first expression

\[

$$\begin{align*} &(x^{2}+8x + 15)-(x^{2}-6x - 9)\\ =&x^{2}+8x + 15 - x^{2}+6x + 9 \end{align*}$$

\]

Step2: Combine like - terms

\[

$$\begin{align*} &(x^{2}-x^{2})+(8x + 6x)+(15 + 9)\\ =&14x+24 \end{align*}$$

\]

Step3: Expand the second expression

\[

$$\begin{align*} &(13mn^{2}-2nm^{2}+5n^{2})-(11mn^{2}-2n^{2})\\ =&13mn^{2}-2nm^{2}+5n^{2}-11mn^{2}+2n^{2} \end{align*}$$

\]

Step4: Combine like - terms

\[

$$\begin{align*} &(13mn^{2}-11mn^{2})-2nm^{2}+(5n^{2}+2n^{2})\\ =&2mn^{2}-2nm^{2}+7n^{2} \end{align*}$$

\]

Step5: Expand the third expression

\[

$$\begin{align*} &(9y^{5}-6y^{3})-(6y^{5}-3y^{4}+5y^{3})\\ =&9y^{5}-6y^{3}-6y^{5}+3y^{4}-5y^{3} \end{align*}$$

\]

Step6: Combine like - terms

\[

$$\begin{align*} &(9y^{5}-6y^{5})+3y^{4}+(-6y^{3}-5y^{3})\\ =&3y^{5}+3y^{4}-11y^{3} \end{align*}$$

\]

Step7: Expand the fourth expression

\[

$$\begin{align*} &(10x^{2}-16x + 13)-(10x^{2}-15x - 10)\\ =&10x^{2}-16x + 13-10x^{2}+15x + 10 \end{align*}$$

\]

Step8: Combine like - terms

\[

$$\begin{align*} &(10x^{2}-10x^{2})+(-16x + 15x)+(13 + 10)\\ =&-x + 23 \end{align*}$$

\]

Step9: Calculate profit for question 5

Profit \(P=\text{Revenue}-\text{Cost}\). Given \(\text{Cost}=10x + 120\) and \(\text{Revenue}=15x+75\).
\[

$$\begin{align*} P&=(15x + 75)-(10x + 120)\\ &=15x+75-10x - 120\\ &=(15x-10x)+(75 - 120)\\ &=5x-45 \end{align*}$$

\]

Step10: Calculate profit for question 6

Substitute \(x = 200\) into the profit formula \(P = 5x-45\) from question 5.
\[

$$\begin{align*} P&=5\times200-45\\ &=1000 - 45\\ &=955 \end{align*}$$

\]

Answer:

1. \(14x + 24\)
2. \(2mn^{2}-2nm^{2}+7n^{2}\)
3. \(3y^{5}+3y^{4}-11y^{3}\)
4. \(-x + 23\)
5. \(5x-45\)
6. \(955\)