Question

I multiplied two trinomials and got a 5-term polynomial, how does this show that polynomials are closed under an operation?

Answer 1

Not an actual question @Vallarylynn16
This is more of a theoretical question if I am reading correctly.

Answer 2

Yes

Answer 3

I know that when you add, subtract, multiply a mono/bi/tri/poly nomial you will get the same result(mono/bi/tri/poly nomial, whichever one you did). So I have to prove it here but I don't know how.

Answer 4

I know what it is asking, but not quite sure how to word it.

Try multiplying this out.

\(\large{(x^2+x+1)(x^2+x+1)}\)

Answer 5

Okay

Answer 6

x^4 + x^3 + x^2 + x^3 + x^2 + x + x^2 + x + 1
Combine like terms:
x^4 + 2x^3 + 3x^2 + 2x + 1

Answer 7

Correct, and how many terms total do you have?

Answer 8

5

Answer 9

This is exactly my problem D:

Answer 10

Okay, so this fits your problem. Now to explain how they are "closed" under an operation.

Answer 11

Yes.

Answer 12

Polynomials, like integers, are "closed" when it comes to addition, subtraction, and multiplication. Basically, this just means they're kind of cliquey as far as these operations are concerned.

An integer plus an integer is an integer, an integer minus an integer is an integer, and an integer times an integer is an integer. Similarly, a polynomial plus a polynomial is a polynomial, a polynomial minus a polynomial is a polynomial, and a polynomial times a polynomial is a polynomial. If that isn't cliquey, we don't know what is.

A polynomial is any expression that is a combination of more than one term via addition or subtraction. Each individual term is called a monomial. Monomials can be constant (like single numbers) or include variables to different degrees (like \(x^6\)). As long as it's in one lump with no plus or minus signs, it's a monomial.

Some examples of polynomials include:

\(x + 4\)
\(x^2 + 2\)
\(2x^2 – 3x + 5\)
\(x^6 + 4x^5 – 3x^2 + x\)

Polynomials are really nice to work with because they're continuous and defined for all values. In other words, we can replace \(x\) with any real number, and we'll get a real number as our result.

For example, take the polynomial \(x^3 + 2x – 5\). Input any value of \(x\), like 6, and we'll get a real number, like \((6)3 + 2(6) – 5 = 223\). You should know that adding, subtracting, and multiplying two or more polynomials together will give them a polynomial. A different polynomial, but still a polynomial.

However, polynomials are not closed (so they're…open?) under division because sometimes the quotient won't be another polynomial. Take this quotient of polynomials, for example.

\(\Large{\frac{x^3+4x^2+2x-5}{x^2+3x+1}}\)

This is a rational expression, not a polynomial. Somehow, polynomials seem a lot more rational.

Definitions are crucial for you to understand before learning how to actually perform operations on polynomials.

You should know that when adding and subtracting polynomials, we can only combine like terms with like terms. Constants can only be added to constants, \(x\) terms can only be added to \(x\) terms, \(x^2\) terms can only be added to \(x^2\) terms, and so on.

If addition and subtraction are like OCD, meticulously pairing terms that go together, then multiplication is like ADHD, combining any and all terms together in one big dog pile regardless of what they are.

You should know that multiplying a polynomial by a monomial means distribution, and that multiplying two polynomials together means a lot of distribution. More specifically, we have to make sure to multiply every term in one polynomial by every term in the other polynomial.

For instance, performing the operation \((x + 2) \times (x^3 + x – 7)\) should know to first distribute the \(x\) to get \(x^4 + x^2 – 7x\), and then the 2 to get \(2x^3 + 2x – 14\), and then to add the two together so that the final answer is \(x^4 + 2x^3 + x^2 – 5x – 14\).

When two binomials are multiplied together, like (x + 1)(x + 3), most prefer to remember the acronym FOIL, which stands for multiplying the First, Outer, Inner, and Last numbers together.

The best way to feel comfortable is to practice, but stick to operations like addition, subtraction, and multiplication at first.



Did this make sense?
I know, it is like a small novel.

Answer 13

On that extended note, I must take my leave. If you need any more help, I am sure that others could probably help :)

Answer 14

But I still don't know the answer to my question!

Answer 15

It means that they have to be grouped with similar terms. Essentially.

Answer 16

In your example I multiplied two trinomials and got 5-terms, which is not a trinomial. How is that an example of the closure property?

Answer 17

I still don't understand..

Answer 18

How do you group 'x^4 + 2x^3 + 3x^2 + 2x + 1' with other terms?

Answer 19

Polynomials are closed under operation when you multiply two trinomials and get a polynomial because it has to be grouped with other terms, essentially.

Is this an answer to that question?

Answer 20

If you were to answer my question in a few sentences, what would you respond?