Study Guide

Polynomials - In the Real World

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In the Real World

Scientific Notation

Earth-shattering fact: scientists use math. Some mathematicians even use science. Too bad more of them don't use soap. Ooh, ice burn.

We're not only talking about white-haired dudes and ladies in lab coats. All scientists, from astronomers to chemists to alientologists, play with numbers that are so big or so small that they had to to think of a way to write them without their hands getting sore. Their hands are already sore enough from having to sign so many autographs for young fans.

The solution? Scientific notation, which is shorthand for writing really big or really small numbers. This is the notation used by movie producers when writing out the amount on one of Johnny Depp's paychecks.

It's called scientific notation because it is useful to all different types of scientist—astronomers use it to measure the size of the universe (big) and chemists use it measure atoms (small)—but even non-scientists such as yourself are allowed to use it. What do you say to the nice scientists? Thank you, scientists.

If you're rusty on decimal multiplication, it would be a good idea to review...oh, about now.

Sample Problem

What is 3,400,000 in scientific notation?

Instead of writing 3,400,000, we could write 3.4 × 106. Using scientific notation in this instance may not be saving us a ton of work, but wait until the numbers become incredibly huge. Hopefully, they'll still remember all the little numbers.

For example, 34,000,000,000,000 is a ridiculously big number. If it wore pants, it would need to shop at a Big and Tall store. It has twelve—count them, twelve—zeros. In order to avoid writing all those zeros, we abbreviate this insanely big number as 3.4 × 1013. If we start with the number 3.4 and move the decimal point 13 places to the right, we find 34,000,000,000,000. If we moved the decimal any more to the right, it would need to change its zip code.

Sample Problem

When numbers become incredibly tiny, negative exponents have a chance to make themselves useful.

We know that 0.000003 is the same thing as . If we start with 3 and move the decimal point 6 places to the left, we'll have 0.000003. By the way, this decimal point is starting to become a little annoyed with all of the back and forth. In fact, it's starting to develop severe motion sickness.

Since we can write  as 10-6,here's how we write 0.000003:

0.000003 = 3 × 10-6

A number like 0.0000000000123 can also be written as 1.23 × 10-11. If we start with the number 1.23 and move the decimal point 11 places to the left, we find 0.0000000000123.

To recap, a number is written in scientific notation if it's in this form:

(number with one value before the decimal place) × 10some integer

Here are a couple more examples:

-24 = -2.4 × 101

3800 = 3.8 × 103

Our first number (before the multiplication sign) always needs to have only one digit before the decimal point. So something like 44 × 105 isn't legit scientific notation. Switch that thing to 4.4 × 106.

Sometimes other symbols besides "×" are used for scientific notation. You may see 123,000 written as 1.23 × 105 or even 1.23 E5. Remember that time you sat on your calculator and that weird number with the "E" appeared? Mystery explained.

When Should Scientific Notation Be Used?

When your teacher or a problem tells you to use scientific notation, it's probably a good idea to follow instructions. If you start questioning authority for no reason, you'll be headed down a slippery slope of juvenile delinquency. You don't want to be sentenced to 1.02 × 103 detentions.

When it's up to you, go with whichever kind of notation makes your hand the least tired. You don't need to use scientific notation for things like "3 + 4 = 7," but you might want to for things like "3,000,000,000,000 + 4,000,000,000,000 = 7,000,000,000,000." Hm, someone must be adding up the scores for Kevin Durant's last three games.

Calculations with Scientific Notation

To do calculations with numbers written in scientific notation without having to turn the numbers back into regular numbers, we use the rules of exponents. A rule of exponents is like a rule of thumb, but with more digits. Ooh, double meaning!

Sample Problem

Find (4 × 103) + (8 × 103). Write the answer in scientific notation.

First we rewrite, using the distributive law, to find (4 + 8) × 103. Then we simplify to 12 × 103. This isn't quite in scientific notation, since the number 12 has two values before the decimal point, so we adjust to get 1.2 × 104 as our final answer. We're making that poor decimal point dizzy again.

Sample Problem

Find (4 × 10-3) ÷ (2 × 102). Write the answer in scientific notation.

Division by 2 × 102 is the same as multiplication by , or . Therefore:

(4 × 10-3) ÷ (2 × 102) = (4 × 10-3)(12 × 10-2)

And we can rearrange things a bit to solve:

This answer is already in scientific notation, so we're done. Now we have a bit of free time to catch up on all our DVR'd episodes of Cake Boss.

  • I Like Abstract Stuff; Why Should I Care?

    Abstract math makes some people flee in horror, but it's actually cool if you give it a chance. We're not saying you need to invite it to your birthday party or anything, but maybe you don't need to lie awake shaking at night because you're fearful it's in your closet. That's a baseless fear, since abstract math is notoriously averse to confined spaces. Hm. Now we're not sure what's more abstract: abstract math, or this paragraph?

    When we put off division until the next unit, we talked about how integers are similar to polynomials. Integers form what's called a ring. You must take this ring into Mordor, and drop it into the top of Mount Doom. Wait, that's something else.

    Here's what a ring means as far as integers are concerned:

    1. Addition is associative, meaning it doesn't matter how you group things: (2 + 3) + 4 = 2 (3 + 4).
    2. Addition is commutative, meaning you can order the numbers any which way: 2 + 3 = 3 + 2.
    3. Zero is a magic number. For every integer m, adding 0 to m leaves m unchanged.
    4. Every integer m has an additive inverse -m, such that m + (-m) = 0.
    5. The distributive law holds.
    6. Multiplication, therefore, is associative.

    How about that? We're back at associative, where we started. That process was rather ring-like, wouldn't you say?

    Polynomials in one variable with integer coefficients also form a ring. We can do a lot of similar things with integers and polynomials: factor, look for primes, look at all the multiples of a certain number or polynomial, whine and complain about them to our algebra teacher, and so on.

    There are many other kinds of rings. The things in a ring are called elements (1 is an element of the ring of integers; x + 3 is an element of the ring of polynomials, etc.).

    Rings and their elements are studied in abstract algebra, which is like regular algebra, only weirder. Tough to imagine, we know. Some rings don't have a multiplicative identity (that is, no number "1"), and some rings have non-commutative multiplication (ab and ba might be different). The other rings only want to be loved. Is that so wrong?

    For more about rings, check this out. For more about The Ring of Power, go here.

  • How to Solve a Math Problem

    There are three steps to solving a math problem.

    1. Figure out what the problem is asking.

    2. Solve the problem.

    3. 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 ft2. 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 (2w), or l = 2w – 1.

    The area of the lawn is given as 15 ft2, so lw = 15. Let's plug our new equation l = 2w – 1 into that area formula.

    (2w – 1)w = 15
    2w2w – 15 = 0

    Hey, lookie there: it's a polynomial equation. A quadratic, to be specific. We can solve the quadratic equation by factoring:

    (2w + 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 width, we find that w = 3 ft and l = 2(3) – 1 = 5 ft. The length of the path is the perimeter of the lawn:

    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. They're planning to drive separately and meet up at the theater. Whatever, saving on gas is for plebes. Serena starts out first, driving at 40 mph, while Blair starts driving 10 minutes later at 50 mph. 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:

    We can rearrange that a bit to get:

    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

    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 + 3q. If he charges $3.50 per box, 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) = 3.5q. Note that R(q) and C(q) are both polynomials of degree 1. The formula for loss is given by subtracting the cost C(q) from the revenue R(q):

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

    The loss is 0 when 0.5q – 800 = 0, which means q = 800 ÷ 0.5 = 1600 boxes.

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

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