Question

The System of PolynomialsYou are aware of the different types of numbers: natural numbers, integers, rational numbers, and real numbers. Now you will work with a property of the number system called the closure property. A set of numbers is closed for a specific mathematical operation if you can perform the operation on any two elements in the set and always get a result that is an element of the set.Consider the set of natural numbers. When you add two natural numbers, you will always get a natural number. For example, 3 + 4 = 7. So, the set of natural numbers is said to be closed under the operation of addition.Similarly, adding two integers or two rational numbers or two real numbers always produces an integer, or rational number, or a real number, respectively. So, all the systems of numbers are closed under the operation of addition.Think of polynomials as a system. For each of the following operations, determine whether the system is closed under the operation. In each case, explain why it is closed or provide an example showing that it isn’t.1)AdditionType your response here:2)SubtractionType your response here:3)MultiplicationType your response here:4)DivisionType your response here:5)Determine whether the systems of natural numbers, integers, rational numbers, irrational numbers, and real numbers are closed or not closed for addition, subtraction, multiplication, and division.Type your response here: 6)Addition Subtraction Multiplication Division natural numbers integers rational numbers irrational numbers real numbers When a rational and an irrational number are added, is the sum rational or irrational? Explain.Type your response here:7)When a nonzero rational and an irrational number are multiplied, is the product rational or irrational? Explain.Type your response here:8)Which system of numbers is most similar to the system of polynomials?Type your response here:9)For each of the operations—addition, subtraction, multiplication, and division—determine whether the set of polynomials of order 0 or 1 is closed or not closed. Consider any two polynomials of degree 0 or 1.Type your response here:10)Polynomial 1 Polynomial 2 Operation Expression Result Degree of Resultant Polynomial Conclusion addition subtraction multiplication division What operations would the set of quadratics be closed under? For each operation, explain why it is closed or provide an example showing that it isn’t.Type your response here:11)Is there a set of expressions that would be closed under all four operations? Explain.Type your response here:

Answer 1

The Solution To Question Number 10:

The question says what operations would the set of quadratics be closed under.

Let the sets of quadratics be

`\begin{gathered} p(x)=ax^2+bx+c \\ q(x)=mx^2+nx+k \end{gathered}`

The set of two quadratics (polynomials) is closed under Addition.

Step-by-step explanation:

`\begin{gathered} P(x)+q(x)=(ax^2+bx+c)+(mx^2+nx+k) \\ =(a+m)x^2+(b+n)x+(c+k) \\ \text{which is still a quadratic.} \\ \text{Hence, the set of quadratics is closed under Addition.} \end{gathered}`

The set of two quadratics is closed under Subtraction.

`\begin{gathered} P(x)-q(x)=(ax^2+bx+c)-(mx^2+nx+k) \\ =(a-m)x^2+(b-n)x+(c-k) \\ \text{which is still a quadratic, provided both a}\\e m,\text{ b}\\e n\text{ } \\ \text{Hence, the set of quadratics is closed under Subtraction.} \end{gathered}`

The set of quadratics is not closed under Multiplication.

`\begin{gathered} P(x)\text{.q(x)}=(ax^2+bx+c)(mx^2+nx+k)=amx^4+(bn+ak)x^2+ck+\cdots \\ \text{Which is not a quadratic.} \\ \text{Hence, the set of quadratics is not closed under multiplication.} \end{gathered}`

The set of quadratics is not closed under Division.

`\begin{gathered} \text{Let the sets be f(x)=8x}^2\text{ and} \\ h(x)=2x^2-1 \\ \text{ So,} \\ (f(x))/(h(x))=\frac{8x^2}{2x^2_{}-1} \\ \text{Which is not a quadratic.} \\ \text{Hence, the set is not closed under Division.} \end{gathered}`