Polynomial Division and Rational Expressions
Math is about translating English into funny symbols. Not laugh out loud funny, usually, but some of them should at least bring a wry smile to your face. What English statements translate into polynomials and rational expressions?
Polynomials often appear in problems where one quantity depends on another. In other words, the quantities are clingy.
Janna has finished weaving a blanket. She wanted the length of the blanket to be 1 foot greater than twice its width, because otherwise her toes get cold. If the area covered by the blanket is 28 square feet, how long is the blanket?
Whenever it makes sense to do so, draw a picture. You can draw a picture when it doesn't make sense to do so, but it's rare that a doodle of Mickey Mouse will help you solve a word problem, unless you're asked to find the circumference of his ears.
We need a variable somewhere. We pick the variable to be the width since the problem discusses the length in relation to the width.
The area covered by this blanket is
length of blanket × width of blanket = (2x + 1)(x) = 2x2 + x.
Since we're told the area covered by the blanket is 28 square feet, we know that
2x2 + x = 28.
To solve this equation, we need to rearrange the terms so that we have a polynomial set equal to 0:
2x2 + x – 28 = 0.
Then we factor:
(2x – 7 )(x + 4) = 0.
The two solutions to this equation are x = 3.5 and x = -4. Do these solutions make sense? x = -4 sure doesn't make sense, because we can't have a blanket that's -4 feet wide. That won't cover even one of Janna's toes. The blanket must be 3.5 feet wide.
Wait, we hope you wrote that down in pencil. We need to make sure we are answering the right problem. We're supposed to give the length, not the width, of the blanket. (Slaps forehead with palm.) To find the length, we plug the width 3.5 in for x in the expression 2x + 1. The length of the blanket is
2(3.5) + 1 = 8 feet.
We can easily check our answers by multiplying the width and length of the blanket, and seeing if we do end up with 28.
3.5 × 8 = 28
Yup, we did it right.
Be careful: When solving word problems involving polynomials or rational expressions, make sure that you only keep those solutions that make sense in the context of the word problem. This often means throwing out one or more negative solutions. They're only bringing everybody down anyway.
There are a few English words and phrases that are frequently used to indicate rational expressions.
A ratio is a comparison between two quantities. We could say "the ratio of books to movies in this house is 3 to 1." Therefore, there are 3 books for every movie. Apparently, the person who was quoted as saying this lived in the 1940's.
A ratio can also be written using a colon (3:1) or using fraction notation ().
That last bit with the fraction notation is how we find rational expressions. And you thought the stork brought them.
The ratio of girls to boys at Maria's school is 2:3, which could either be a good or a bad thing for Maria, depending on who you ask. If there are 22 girls at Maria's school, how many boys are there?
The ratio of girls to boys is . It is also , where x is the number of boys, so .
To solve this equation we multiply both sides by 3 and then both sides by x to find 2x = 66. x = 33, so there are 33 boys at Maria's school, and not one of them knows how to treat a lady.
That example was also an instance of a proportion, which is an equation that says two ratios are equal. It is "pro-equal portions," if you want to think of it that way. If you don't, that's cool, too. The statement
is a proportion.
If Liana needs $5 to buy 6 pencils, how much money does she need to buy 7 pencils? Also, what in the world is she doing spending so much for pencils? Were these famous pencils?
We can set this up as a proportion:
where x is the amount of money Liana needs, assuming she hasn't come to her senses and picked up a $1.49 12-pack from Office Depot.
Multiplying both sides by 7 gives .
Since , Liana would need $5.84 to buy 7 pencils.
Another idea that appears around rational expressions is the idea of proportionality. We say y is directly proportional to x if
y = (some constant) × x
The constant is called the constant of proportionality. Wherever did they come up with that one?
If y = 5x, then y is directly proportional to x with constant of proportionality 5.
If y is directly proportional to x, it means two things. Or at least two things, but these are our favorites. We have a list somewhere with a ranking of our top four hundred, if you'd like to see it.
- As the magnitude of x becomes bigger, so does the magnitude of y. If x gets farther from zero, so does y, since y is x multiplied by something. In other words, x and y are directly related...which is not to say they don't avoid each other at family functions.
- The ratio of y to x is always the same. will always be equal to the constant of proportionality (if y = 5x, then is always 5).
Assume that Tina walks at a constant rate. In other words, she doesn't stop along the way to pick up any pennies or caterpillars. The distance she travels is directly proportional to the time she walks, where the proportionality constant is the rate at which she walks. If Tina travels 3.5 miles over an hour and fifteen minutes, how fast did she walk? Follow-up question: how much money could she have had if she had picked up all those pennies?
Answer. There are so many words here, including many that are utterly silly, they're obscuring what's going on. Let's translate into math a little at a time. Someone summon the interpreter.
We're told "the distance she travels is directly proportional to the time she walks," which means
distance = constant × time.
Since we're told the proportionality constant is her rate, we use this equation, which may already be familiar to you:
distance = rate × time.
Bells should be going off in your head, and not the ones you usually hear whenever you look into a bright light. You should see a doctor about that, by the way. Tina's distance was miles and she walked for hours, so
We can solve this by eliminating the denominators. Multiply both sides by 4 to find
14 = (rate) × 5,
so her rate was miles per hour. Hey Tina, where's the fire?
For the sake of completeness, there's also something called inverse proportionality. Not to be confused with Converse proportionality, which makes sure that both of your shoes are the same size. We say y is inversely proportional to x if
If then y is inversely proportional to x with constant 7.
If y is inversely proportional to x, then
- As the magnitude of x becomes bigger, the magnitude of y becomes smaller.
- xy is always equal to the constant (if , then xy is always 7.)