polynomial word problems

[laurakate]

Suppose a bank makes a loan for the principle amount P and charges interest at the rate r. Write an expression for the amount of interest owed to the bank at the end of one year. If no payment of principle or interest is made at the end of the first year but the loan is extended for a second year at the same rate write an expression for the principle amount of the second year's loan. Write an expression for interest owed at the end of the second year? Write an expression. Give a complete factorization for the polynomial (in r) of last question.

 

[mmm4444bot]

Please read the post titled, "Read Before Posting".

It outlines your responsibilities for seeking help on these boards.

Namely, we don't do your homework, and you need to start making some statements about what you don't understand, OR ask some specific questions.

I did provide you with some help on your other post, but I had to guess at what to say because you didn't ask any questions in that post, either.

What formulas have you learned for dealing with simple interest? Do you know what the words "principal" and "interest" mean?

Thank you.

 

[laurakate]

My apologies for not posting what exactly I needed help on!

I have tried working this problem out and thinking it through and I got as far as A, but am not sure it's correct. The expression I got for the amount of interest owed to the bank at the end of one year was: 1/r=P + Pt1/a.

For the expression for the principle amount of 2nd year's loan would I just square the P's in the above formula?

For the expression for interest owed would I square the r instead of the P's?

To find the total amount owed at the end of the second year, which expression am I supposed to use? And I will just be simplifying because these are not possible to solve, correct?

To give a complete factorization for the polynomial (in r) of part d.....I don't even understand what this is asking me to factor! Can you explain to me what it is talking about?

Thanks!

 

[mmm4444bot]

I have tried working this problem out and thinking it through and I got as far as A What is A?

The expression I got for the amount of interest owed to the bank at the end of one year was: 1/r=P + Pt1/a.

This is not an expression; it's an equation. (I'm not sure what you started with, or what you did to obtain this incorrect result, because you did not show your work.)

For the expression for the principle amount of 2nd year's loan would I just square the P's in the above formula?

No. The balance owed at the end of year 1 becomes the new principal at the beginning of year 2.

For the expression for interest owed would I square the r instead of the P's? No. Use the formula for simple interest.

To find the total amount owed at the end of the second year, which expression am I supposed to use?

Again, use the formula for simple interest.

And I will just be simplifying because these are not possible to solve, correct? Yes, everything is symbolic, in this exercise, because we don't have any of the actual numbers.

To give a complete factorization for the polynomial (in r) of part d.....I don't even understand what this is asking me to factor! Can you explain to me what it is talking about?

The expression for the balance owed at the end of the second year is an expression containing the symbols P, r and t. They're just asking you to factor this expression.

I'm thinking that this exercise is intended as a prelude to teaching you about compounded interest. Basically, if the principal+interest is reinvested by the bank for a second year, then interest is going to accumulate on the first year's interest, in addition to the original principal. That's compounding -- interest earned on interest.)

You seem totally confused, so I'll try to walk you through it.

Here is the formula for calculating simple interest (I) on principal P at interest rate r for t years.

I = P * r * t

In other words, with simple interest, we just multiply the principal times the decimal form of the interest rate times the number of years. That gives us the interest only.

Here' an example, using Real numbers.

I borrow $1,000 for 1 year at interest rate 5%.

I = 1000 * 0.05 * 1

I = 50

The bank charges $50 dollars interest for that year. I have to pay back both the principal and the interest, so the amount that I owe the bank after 1 year is $1050.

Now, if the bank had agreed to loan me the $1,000 for two years, instead (NOT the scenario in your exercise; just an example), and the bank agreed to charge only simple interest, then I would only have to pay interest on $1,000 for each of the two years.

I = 1000 * 0.05 * 2

I = 100

After two years, I owe the bank $1,100 on a simple interest loan.

Again, this is not the scenario, in your exercise! In your exercise, the bank is treating the two years as two different loans, actually.

In your exercise, my example goes like this:

At the end of year 1, I owe $1,050. The bank then extends me another year's loan under the same terms, but the principal is now $1,050 (what I currently owe), instead of $1,000 (what I initially borrowed).

I = 1050 * 0.05 * 1

I = 52.50

See how the interest charged on the second year is a little more than $50? That's because the bank charged interest on not only the initial $1,000, but also on the $50 that I owed from year 1. (That's compounding -- interest charged on interest.)

So, after 2 years, the total owed under this scheme is $1,102.50 .

Okay, back to the symbolic forms.

Since the borrower has to pay back both the borrowed principal P and the simple interest charged Prt, we write the expression for the amount owed after t years as follows.

P + Prt

Now, what is this after one year?

If we're talking about 1 year, then t = 1, right?

P + Pr

So, the expression P + Pr is the amount owed at the and of year 1. (It's the sum of the principal P plus the interest Pr.) We can factor this expression.

P(1 + r)

This entire amount is then treated as the new principal at the beginning of year 2.

First, I'll make a substitution, so the expression won't be so daunting; then I'll reverse the subtitution, to show you what this exercise expects to see for the second year (i.e., the daunting version, heh, heh).

Let A = P(1 + r)

So, in my example above with Real numbers, A = 1050.

Now, the bank starts over with this new principal A, for year 2.

The interest charged for year 2 is calculated using the very same formula for simple interest.

I = A * r * t

t = 1, again, right?

I = Ar

Again, this is only the interest. We need to pay back both the interest Ar plus the principal A.

Therefore, at the end of year 2, we owe the following amount.

A + Ar

Now, I reverse the subsitution [i.e., I replace A with the original expression from year 1: P(1 + r) ].

P(1 + r) + P(1 + r)r

Since the expression P(1 + r) appears on both sides of the red plus sign, we can factor it out.

P(1 + r) * (1 + r)

P(1 + r)^2

This is the amount that's owed to the bank at the end of year 2.

This is compounded interest now, because interest was charged (in year 2) on interest from year 1.

So, we have the following formula for compounded interest, when the interest is compounded once per year:

A = P(1 + r)^t

Let's see what happens to the amount owed after two years (A) using this formula with my Real-numbered example, instead of using the two-step process described n your exercise.

A = 1000(1 + 0.05)^2

A = 1102.50

See? It works out to be the same amount. That's the benefit to the annually compounded formula. It allows us to do the calculations "all at once", without having to do them year-by-year.

Questions? (This is a long response, so I might have made errors. Read carefully.)

 

[laurakate]

This is actually a question in chapter 6 which covered rational expressions and the next chapter we do covers radical expression and equations and quadratic equations!

Your explanation really helped. So Principle (interest + rate) tells the amount of interest owed at the end of one year. Principle being the money invested, interest being what you pay on top of the loan so the bank makes money, and rate being the rate the bank gives.

If no payment is made at the end of the first year and loan is extended for a second year at the same rate, how do you know that it automatically covers the amounts that were to be paid in the first year? This expression is P+(1+r)^2? I kind of got confused with the formulas!

An expression for interest owed at end of year 2 of the loan would be I=Prt. t being 2...

Total amount owed at end of second year is: A=P(1+r)^2. Which means principle times (1+rate)squared.

Complete factorization for polynomial A=P(1+r)^2 is P(1+r)(1+r)


You are correct in saying I'm completely lost and totally confused. I understand what principle, rate, interest, and time are and where the numbers come from...but I had no idea about compounded interest...and i don't know the formula for simple interst. So far you've shed more light than my text, my teacher, and another program we have access to! Thanks soooo much!

 

[Mrspi]

You are correct in saying I'm completely lost and totally confused. I understand what principle, rate, interest, and time are and where the numbers come from...but I had no idea about compounded interest...and i don't know the formula for simple interst. So far you've shed more light than my text, my teacher, and another program we have access to! Thanks soooo much!

If you don't know the formula for "simple interest," then the previous response probably doesn't mean much to you.

You need to start with the simple interest formula:

Interest = Principal * interest rate in decimal form * time in years

If that doesn't make sense to you, then please talk to your teacher.

Once you've got that concept down, please look carefully at the excellent responses you've received here.

 

[laurakate]

Thank y'all. I truly appreciate your help and your efforts!

 

[mmm4444bot]

This is actually a question in chapter 6 which covered rational expressions That's odd.

Your explanation really helped. So Principle (interest + rate) tells the amount of interest owed at the end of one year.

Nope.

We can't write "interest + rate".

"interest" is money. "rate" is not money. We can only add money to money.

The interest owed at the end of one year is expressed using the formula that we've provided you.