#### CHECK OUT SHMOOP'S FREE STUDY TOOLS: Essay Lab | Math Shack | Videos

# Polynomials

# How to Solve a Math Problem

There are three steps to solving a math problem.

- Figure out what the problem is asking.
- Solve the problem.
- Check the answer.

### Sample Problem

It's early spring, and Cierra is planting a new lawn. She's decided that the length of the lawn should be 1 ft less than double the width, and the area should be 15 ft^{2}. She also plans to construct a concrete path around the lawn. Fancy. Find the length of the path.

In this problem, we need to find the perimeter of the lawn. Since the dimensions of the lawn are unknown, let *l* be the length and *w* be the width. The length (*l*) is 1 ft less than double the width (2*w*), or *l* = 2*w* – 1.

The area of the lawn is given as 15 ft^{2}, so

*lw* = 15

(2*w* – 1)*w* = 15

2*w*^{2} – *w* – 15 = 0.

We can solve the quadratic equation by factoring

(2*w* + 5)(*w* – 3) = 0, so and 3.

After discarding the negative value, since it would be extra-super-fancy (not to mention bizarre) to have a negative lawn, *w* = 3 ft and *l* = 2(3) – 1 = 5 ft. The length of the path is 2(*l* + *w*) = 2(5 + 3) = 16 ft.

### Sample Problem

Serena and her BFF (or for this week, anyway) Blair want to go to a movie playing at a theater 45 miles away. Whatever, saving on gas is for plebes. Serena starts out first, driving at 40 miles/hour, while Blair starts driving 10 minutes later at 50 m/hr. When will Blair pass Serena?

(Because she *will* pass Serena. Hey, nobody ever said they were great drivers.)

Let *t* be the time in hours when Blair passes Serena. Therefore, the distance traveled by Blair in *t* hours will be same as the distance traveled by Serena in *t* hours and 10 minutes (she starts driving 10 minutes, or hr, earlier). Remember the speed formula:

implies distance = speed × time.

Using Serena and Blair's driving speed given in the problem, we solve the following linear polynomial equation:

Blair will pass Serena after 40 minutes. Eat her dust, xoxo!

### Sample Problem

Lisa Simpson is planning to deposit her pocket money in the bank, like the bizarrely responsible eight-year-old she is. If she is depositing $500 and the interest rate is *r*% per year, how much money will she have in her account after 2 years at the ripe old age of 10?

Using simple interest formula, Lisa will have after one year.

After two years, she will have

,

which is a polynomial of degree 2 in *r*. Just in time for fifth grade, too.

### Sample Problem

Stewie stole $5 from Brian and has been trying to hide away. Not because he's scared, just because of...reasons. Brian goes looking for him running north at 4 miles/hour, so Stewie starts running (okay, toddling) east at 3 miles/hour. When will they be separated by 1 mile?

Let *t* be the time when Stewie and Brian are 1 mile apart. In *t* hours, Brian has traveled 5*t* miles and Stewie has traveled 3*t* miles. Using the Pythagorean formula, we get the quadratic equation:

They will be a mile apart in 12 minutes.

### Sample Problem

Bobby Hill decides to open a factory to produce Shmickerdoodle cookies. He found out that the cost of making *q* boxes is given by *C*(*q*) = 800 + 3*q*. If the cost of each box is 3.50, how many boxes should the factory produce in order to have no loss?

The revenue obtained by selling *q* boxes is the price of each box times the number of boxes sold, *R*(*q*) = 3.5*q*. Note that *R*(*q*) and *C*(*q*) are both polynomials of degree 1. The formula for loss is given by subtracting revenue *R*(*q*) from the cost *C*(*q*):

Loss = *R*(*q*) – *C*(*q*) = 800 + 3*q* – 3.5*q* = 800 – 0.5*q*.

The loss is 0 when

That's a long way from breaking even, Bobby-o. Hope those Shmickerdoodles are worth it.