Precalculus Write the following expressions of the form (a+bx)^n

[PeachBlossom]

I know how to approach problems with expanding expression with bionomial theorem and expanding expressions with more then 2 terms with bionomial theorem.

But now I have a problem. Now I need to do the completely opposite. I need to condense the expression to this form (a+bx)^n

How do I condense it? Is there a specific method or a formula I need to use. Please help.

I have no idea to do this, no information about this either in my books.

Thanks in advance!

 

[Steven G]

Hmmm, I thought in your last post you said that you know how to expand (a+b)ⁿ using binomial expansion. This problem is exactly the same except that b is replaced with bx. Just do the as with (a+b)ⁿ but again, it not b this time, it is bx.
In part a, n=3 and in part b, n=4. Do you see that?
Please show us your work!

 

[PeachBlossom]

Im sorry for being very confused now. But on this question isn't it completely opposite than I have done.

Because the question looks already done after you have done binomial theorem. And now I need to reverse the process and go back.

Do you understand my thought process? Tell me if it is wrong so I don't stick this in my memory.

 

[Dr.Peterson]

I know how to approach problems with expanding expression with bionomial theorem and expanding expressions with more then 2 terms with bionomial theorem.

But now I have a problem. Now I need to do the completely opposite. I need to condense the expression to this form (a+bx)^n

How do I condense it? Is there a specific method or a formula I need to use. Please help.

I have no idea to do this, no information about this either in my books.

Thanks in advance!

View attachment 14227

Yes, this is the opposite; but it's easier, assuming you really understand the expansion.

Do you see that when you expand [MATH](a+bx)^n[/MATH], the first term of the expanded form will be [MATH]a^n[/MATH], and the last will be [MATH]b^nx^n[/MATH]?

So if the last term is [MATH]x^3[/MATH], what must [MATH]n[/MATH] be? What must [MATH]b[/MATH] be?

And then, if the first term is 8, what must [MATH]a[/MATH] be?

This allows you to quickly guess what the answer has to be, if it can be done at all. Then you just have to expand your answer to make sure it is right.

 

[Steven G]

Personally I find that technique dangerous as there might be a mistake in the exercise (ie (2+x)³ may not actually equal (8 +12x +6x² + x³) ) and one would not catch it using your method.

Surely if this was on an SAT type exam I would 100% use your method. I am just the type of person who just likes to confirm that the choice is really the correct answer and that the problem is not flawed.

 

[PeachBlossom]

Could you show your technique that is safer? Would really appreciate it.

 

[Steven G]

I would expand (a+bx)³ and equate the coefficients with 8 +12x +6x² + x³

However, at your stage right now I would use Dr Peterson's method. To be honest my post was just my opinion from someone who has a bit more experience than you. I apologize for confusing you with my post.

 

[JeffM]

When a teacher or test asks a question like "Reduce this polynomial to the form (a + bx)^n," it is a reasonable supposition that the polynomial is indeed an expansion of (a + bx)^n.

So as a tactic for getting your homework done in a hurry or for answering as many questions correctly as possible on a timed test, Dr. Peterson is giving truly excellent advice. Moreover, he gives the important caveat to test the result.

Unfortunatly, an infinite number of polynomials cannot be simplified to the form (a + bx)^n. Questions like this induce students to infer that a special case is a general case.

My objection is not to Dr. Peterson's advice but rather to the nature of the question.

The question should be worded

Indicate whether the following polynomial can be reduced to the form of (a + bx)^n and, if it can, what are a and b.

Worded that way, no false inference is possible, and the danger, if in fact that is an apt charactetization, in Dr. Peterson's answer is eliminated.

 

[pka]

I know how to approach problems with expanding expression with bionomial theorem and expanding expressions with more then 2 terms with bionomial theorem.

But now I have a problem. Now I need to do the completely opposite. I need to condense the expression to this form (a+bx)^n

How do I condense it? Is there a specific method or a formula I need to use. Please help.

I have no idea to do this, no information about this either in my books.

Thanks in advance!

View attachment 14227

Look at this link.
You need to learn to use the on-line resources like that one.
Can you guess what part (b) is and then use the link to show it?

 

[Steven G]

Yes, I 100% would show a student doing homework or preparing for a test the same method as Dr Peterson stated. In fact, in my post to the OP I suggested that s/he uses Dr Peterson's method! I am no longer a student and just feel more comfortable making sure all is correct. I even apologized to the student for confusing them. Again, it is just a habit I acquired. To point out why I have this habit just consider the recent post where the problem was something like _48*3_7 = 6_5 (which has no solution).
I guess that I should not have posted my opinion/habit here. You are right about that. And I certainly think that (for a student) Dr P's method was excellent.

 

[Dr.Peterson]

A similar, but simpler, problem is "Factor the following as a perfect square". Such a question presumes that it can be done; so it is useless in real life, where you would never have reason to do it when you run across a random polynomial without a teacher giving you instructions ...

Except that sometimes you do! Given a trinomial like 64x^2 - 176x + 121, if the goal is to factor it, you have a lot of work ahead of you if it is a typical trinomial; but you can wonder if it just might happen to be a perfect square, since the first and last terms are perfect squares. So you try it: If it is a perfect square, then it must be the square of (8x - 11), in order to get the first and last terms, and the middle sign, right. So you make a hopeful guess and check it. It is, so you've saved all that work (with no risk at all, since of course you knew enough to check, because you had no reason to assume it is a perfect square -- you just hoped it was).

So this sort of thing is a good strategy, with no risk if you are thinking correctly. As for the OP, if the check failed, you'd just have to say "It can't be done" -- just as the solution to some equations is "no solution". But I agree that the problem ought to say that.

By the way, in writing this, I realized I didn't say everything you have to think about in problem 9(b); there might actually be two possibilities to check, with different signs. It wasn't my intent to give a complete method, just some hints.

Now, if you were to equate coefficients, you would get 4 (or 5) equations in two unknowns, and some students might still forget to check for consistency with that method as well. In effect, my approach just writes the two easiest of the equations, and then promises to check the answer by a slightly easier method.

 

[PeachBlossom]

Thank you for all the help everyone appreciate it.

I understood and I think I solved the problem.

Is this an acceptable answer? My teacher want us to show how we solved the problem with allot of writing.

 

[HallsofIvy]

This problem is much easier than that because you are told that the polynomial is of the form "\(\displaystyle (a+ bx)^n\)"! For (a) the first thing I would notice is that the highest power of x is \(\displaystyle x^3\) and its coefficient is 1 so I know immediately that n= 3 and b= 1. I would then notice that the term that does NOT include x is \(\displaystyle 8= 2^3\) so a is 2.

 

[Dr.Peterson]

What you've written is the long way I mentioned, with four equations; it is correct, and would be what a teacher would want if they require you to do unnecessary work. What Halls said is what I initially suggested, and amounts to looking only at the first and last of your equations.

The second problem involves a little extra thinking (though not much), as I've suggested.

But if the issue is only that you are expected to show work, I would write what Halls did, PLUS expanding your answer to show that every term is correct. I would never solve a problem based on the assumption that the question itself is valid.