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### Determine which operations are closed for polynomials

Determine which operations are closed for polynomials

Determine which operations are closed for polynomials

### MathBitsNotebook(A1) ^{[1]}

||A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.. The leading term in a polynomial is the term of highest degree.

Remember: NO negative exponents! NO fractional exponents!. NO – the square root of x can be written with a fractional exponent of x.

||The degree of a term with whole number exponents is the sum of the exponents of the variables, if there are variables. Non-zero constants have degree 0, and the term zero has no degree

### Common Core Map ^{[2]}

Algebra: Arithmetic with Polynomials and Rational Expressions722 questions44 skills. Algebra: Arithmetic with Polynomials and Rational Expressions

Multiply monomials by polynomials (basic): area model. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Prove polynomial identities and use them to describe numerical relationships.. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

### MathBitsNotebook(A1) ^{[3]}

||A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.. The leading term in a polynomial is the term of highest degree.

Remember: NO negative exponents! NO fractional exponents!. NO – the square root of x can be written with a fractional exponent of x.

||The degree of a term with whole number exponents is the sum of the exponents of the variables, if there are variables. Non-zero constants have degree 0, and the term zero has no degree

### What operations is not closed for polynomials? ^{[4]}

Yes, because there is no way of multiplying two polynomials to get something that isn’t a polynomial.. It means that you can do any of those operations, and again get a number from the set – in this case, a polynomial

In my opinion the question is poorly defined, since “non-polynomial” could be just about anything.. Polynomials are the simplest class of mathematical expressions

Yes, because there is no way of multiplying two polynomials to get something that isn’t a polynomial.

### Which Operation Is NOT Closed For Polynomials? A)add A Trinomial To A Trinomial B)divide A Binomial By ^{[5]}

Since we know that polynomials are closed under addition, subtraction and multiplication but not under division because division of two polynomials is not necessarily a polynomial.. Dividing a binomial by a trinomial will result in a rational expression and it will have negative exponent

If the sale price was 57.75, what was the original price of the stereo?. Since the car stereo system was on sale for 30% off, this means that it was sold for 100% – 30% = 70% of the original price.

Which would not be a step in solving 3x+1+2x=2+4x? A. How much profit would a restaurant make if they sold 1 scrambled egg for 99 cents?

### Closure (mathematics) ^{[6]}

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations

Let S be a set equipped with one or several methods for producing elements of S from other elements of S.[note 1] A subset X of S is said to be closed under these methods, if, when all input elements are in X, then all possible results are also in X. Sometimes, one may also say that X has the closure property.

### Why missing the zero polynomial indicate subset is not closed under multiplication? ^{[7]}

Let $V$ be a vector space over a field $F$ and let $H$ be a subset of $V$. We will say that $H$ is a vector subspace of $V$ $\color{blue}{\text{if, and only if}}$, the following conditions are satisfied:

– For all $\alpha$ in $F$ and for all $h$ in $H$, we have $\alpha\cdot h\in H$.. Now, consider the set $$H=\{a_{0}+a_{1}x+a_{2}x^{2}\in P_{2}(\mathbb{R}): a_{0}+2a_{1}+a_{2}=4\}$$

Thus $H$ will be a vector subspace of $P_{2}(\mathbb{R})$ if, and only if, are satisfied the conditions $1),2)$ and $3)$.. Now, recall that $H$ is a subset of polynomials in $P_{2}(\mathbb{R})$ which satisfies one specific condition, that is, $a_{0}+2a_{1}+a_{2}=4$.

### Sources

- https://mathbitsnotebook.com/Algebra1/Polynomials/POpolys.html#:~:text=Multiplying%20polynomials%20creates%20another%20polynomial,a%20variable%20in%20the%20denominator).
- https://www.khanacademy.org/commoncore/grade-HSA-A-APR#:~:text=Understand%20that%20polynomials%20form%20a,%2C%20subtract%2C%20and%20multiply%20polynomials.
- https://mathbitsnotebook.com/Algebra1/Polynomials/POpolys.html
- https://math.answers.com/algebra/What_operations_is_not_closed_for_polynomials
- https://plataforma.unitepc.edu.bo/answers/2171657-which-operation-is-not-closed-for-polynomials
- https://en.wikipedia.org/wiki/Closure_(mathematics)
- https://math.stackexchange.com/questions/4452077/why-missing-the-zero-polynomial-indicate-subset-is-not-closed-under-multiplicati