To submit Let R be a ring. Define a function e : R[x] × R → R by the rule e(a„x" + ·..+a¡x+a0,r) = a,r" + · · ·+a¡r+ao. In other words, given the input of a polynomial f(x) E R[x] and an element r E R, e(f(x),r) is obtained by substituting r for x in f(x). (a) Prove that, if R is a commutative ring, then e(f,r)+e(g,r) = e(f+g,r) and e(f,r) · e(g,r) = e(f8,r) for any two linear polynomials f,g € R[x] and any r E R. [In fact both facts are true for polynomials f,g of any degree. I encourage you to write up a proof of the more general versions instead, if you can. But I didn’t want to burden everyone with having to use indices and arbitrarily long sums.]
Quadratic Equation
When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.
Demand and Supply Function
The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
Could anyone give me the solution?
Thanks
![To submit Let R be a ring. Define a function e : R[x] × R → R by the rule
e(anx" +...+a¡x+ao,r) = anp" +.…+a¡r+ ao-
In other words, given the input of a polynomial f(x) E R[x] and an element r E R,
e(f(x),r) is obtained by substituting r for x in f(x).
(a) Prove that, if R is a commutative ring, then
e(f,r)+e(8,r) = e(f+g,r) and
e(f,r) · e(g,r) = e(f8,r)
for any two linear polynomials f,g € R[x] and any r E R.
[In fact both facts are true for polynomials ƒ,g of any degree. I encourage you
to write up a proof of the more general versions instead, if you can. But I didn’t
want to burden everyone with having to use indices and arbitrarily long sums.]
(b) If R is a non-commutative ring, just one of the two equations in part (a) still holds
for all r, f, and g. Which one? Explain how you could make a counterexample
to the other equation.
[If you know a non-commutative ring, feel free to give an actual counterexample
in that ring instead of “explaining how".]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b0fc8b5-09e3-47e4-9f8b-17d7c3ed911d%2F1cb8b793-c1df-4c8f-aea7-66252bfcdce6%2Ftvlf71w_processed.png&w=3840&q=75)
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