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A rectangle has sides measuring (4x + 5) units and (3x + 10) units.


Part A: What is the expression that represents the area of the rectangle? Show your work. (4 points)


Part B: What are the degree and classification of the expression obtained in Part A? (3 points)


Part C: How does Part A demonstrate the closure property for polynomials? (3 points)

Sagot :

Part A

Like with any other rectangle, we multiply the length and width to get the area

This applies even when we have algebraic expressions like this (recall that x is simply a placeholder for a number).

So,

area = length*width

area = (4x+5)*(3x+10)

area = w(3x+10) ..... replace (4x+5) with the variable w

area = 3xw + 10w ..... distribute

area = 3x(4x+5)+10(4x+5) .... plug in w = 4x+5

area = 3x*4x+3x*5 + 10*4x+10*5 .... distribute again

area = 12x^2+15x + 40x+50

area = 12x^2+55x+50

Answer: 12x^2+55x+50

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Part B

The degree of any single variable polynomial is the largest exponent. In this case, the largest exponent is 2. This 2nd degree polynomial is considered a quadratic. Because we have three terms, this quadratic polynomial is also a trinomial. So there are many ways to describe the result of part A. Possibly the quickest way to describe it is to say "quadratic trinomial" to say all that has been mentioned in this paragraph.

Answers:

  • Degree = 2
  • Classification = quadratic trinomial

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Part C

The concept of closure is the idea of taking two objects of the same type, applying some operation to them, and getting an output object of the same type.

Admittedly that last paragraph is a bit vague so I'll go over a few examples.

  • If we take any two integers and add them up, then we get another integer. This shows integers are closed under addition.
  • We can also multiply any two whole numbers to get some other whole number. This is another example of closure.

A non-example would be dividing whole numbers. Something like 10/2 = 5 is a whole number but 10/3 = 3.33 is not. This shows that integers are not closed under division. We don't always get a whole number output.

With polynomials, multiplying any two of them will result in some other polynomial. It's basically using the two examples in the bullet points and building upon them. That's the short version of why polynomials are closed under multiplication.

For this particular problem, (4x+5), (3x+10), and 12x^2+55x+50 are all polynomials.  

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In terms of notation,

polynomialA * polynomialB = polynomialC

This notation hopefully shows that multiplying any two polynomials leads to another polynomial.

Refer to this previous problem I answered

https://brainly.com/question/24065851

Specifically, the top of part C talking about a coin machine should hopefully help if any of the concepts here don't make sense. If neither this current page nor the link helps, then let me know and I'll try to rephrase things another way.

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