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Which property of polynomial multiplication says that the product of two polynomials is always a polynomial?
That property is called CLOSURE.
What property of polynomial multiplication says that the product of two polynomials is always a polynomial?
Clouser
Do you use the distributive property with a monomial and a polynomial?
If you want to multiply the monomial by the polynomial, yes. In that case, you have to multiply the monomial by every term of the polynomial. For example: a (b + c + d) = ab + ac + ad More generally, when you multiply together two polynomials, you have to multiply each term in one polynomial by each term of the other polynomial; for example: (a + b)(c + d) = ac + ad + bc + bd All this can be derived from the distributive property (just apply the distributive property repeatedly).
Can multiplication be distributive over subtraction?
Multiplication can be the first step when using the distributive property with subtraction. The distributive law of multiplication over subtraction is that the difference of the subtraction problem and then multiply, or multiply each individual products and then find the difference.
Is subtraction an identity property?
Subtraction is not an identity property but it does have an identity property. The identity is 0 and each number is its own inverse with respect to subtraction. However, this is effectively the same as the inverse property of addition so there is no real need to define it as a separate property.
What property of polynomial subtraction says that the difference of two polynomials is always a polynomial?
The property of polynomial subtraction that ensures the difference of two polynomials is always a polynomial is known as closure under subtraction. This property states that if you take any two polynomials, their difference will also yield a polynomial. This is because subtracting polynomials involves combining like terms, which results in a polynomial expression that adheres to the same structure as the original polynomials.
Which property of polynomial subtraction says that the difference of two polynomials is always a polynomial?
The property that states the difference of two polynomials is always a polynomial is known as the closure property of polynomials. This property indicates that when you subtract one polynomial from another, the result remains within the set of polynomials. This is because polynomial operations (addition, subtraction, and multiplication) preserve the degree and structure of polynomials. Thus, the difference of any two polynomials will also be a polynomial.
Which property of polynomial addition says that the sum of two polynomials is always a polynomial?
It is called the property of "closure".
Which property of polynomial multiplication says that the product of two polynomials is always a polynomial?
That property is called CLOSURE.
What property of polynomial multiplication says that the product of two polynomials is always a polynomial?
Clouser
Division of Monomials and Polynomials by Monomials?
to multiplya polynomial by a monomial,use the distributive property and then combine like terms.
Why are polynomials not closed under division?
Polynomials are not closed under division because dividing one polynomial by another can result in a quotient that is not a polynomial. Specifically, when a polynomial is divided by another polynomial of a higher degree, the result can be a rational function, which includes terms with variables in the denominator. For example, dividing (x^2) by (x) gives (x), a polynomial, but dividing (x) by (x^2) results in (\frac{1}{x}), which is not a polynomial. Thus, the closure property does not hold for polynomial division.
Explain how you multiply two polynomials?
To multiply two polynomials, you apply the distributive property, also known as the FOIL method for binomials. Each term in the first polynomial is multiplied by each term in the second polynomial. After performing all the multiplications, you combine like terms to simplify the resulting polynomial. Finally, ensure that the polynomial is written in standard form, with terms ordered by decreasing degree.
Do you use the distributive property with a monomial and a polynomial?
If you want to multiply the monomial by the polynomial, yes. In that case, you have to multiply the monomial by every term of the polynomial. For example: a (b + c + d) = ab + ac + ad More generally, when you multiply together two polynomials, you have to multiply each term in one polynomial by each term of the other polynomial; for example: (a + b)(c + d) = ac + ad + bc + bd All this can be derived from the distributive property (just apply the distributive property repeatedly).
How are operations and properties of real numbers related to polynomials?
Operations and properties of real numbers, such as addition, subtraction, multiplication, and division, directly apply to polynomials since they are composed of real number coefficients and variables raised to non-negative integer powers. Polynomials can be manipulated using these operations, allowing for the application of properties like the distributive property, the commutative property, and the associative property. Additionally, the behavior of polynomials, including their roots and behavior at infinity, is fundamentally linked to the properties of real numbers. Thus, understanding real number operations is essential for working with and analyzing polynomials.
Can multiplication be distributive over subtraction?
Multiplication can be the first step when using the distributive property with subtraction. The distributive law of multiplication over subtraction is that the difference of the subtraction problem and then multiply, or multiply each individual products and then find the difference.
How does the distributive property get factored into a polynomial?
The distributive property allows us to simplify expressions by distributing a term across a sum or difference. When factoring a polynomial, we can reverse this process by identifying common factors in each term of the polynomial. For example, in the expression ( ax + ay ), we can factor out ( a ) to get ( a(x + y) ). This reveals the common factor and simplifies the polynomial into a product of its factors.
