Question

A rectangle has sides measuring (2x + 7) units and (5x + 9) units.

Part A: What is the expression that represents the area of the rectangle? Show your work.
Part B: What are the degree and classification of the expression obtained in Part A?
Part C: How does Part A demonstrate the closure property for polynomials?

Answer 1

Part A

Use the FOIL rule to expand out the expression below

(2x+7)(5x+9)

2x*5x + 2x*9 + 7*5x + 7*9

10x^2 + 18x + 35x + 63

10x^2 + 53x + 63  is the final answer for part A

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Part B

The degree is 2. This is a quadratic trinomial.

The degree is simply the largest exponent after expanding everything out. The degree directly determines the type of polynomial. Since the degree is 2, we have a quadratic. The polynomial is a trinomial because it has 3 terms.

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Part C

Imagine we had a machine in which we deposit 2 coins into it. One would happen on either side. After the coins are deposited, a third coin pops out. The 2 coins going in and the third coin coming out are the same currency. This represents an example of closure.

In mathematics, the idea of closure is taking any two objects of the same type, applying an operation on them, and getting the same type of object as the input.

For example, we could take two whole numbers and multiply them together. The result is always a whole number. The set of whole numbers is closed under multiplication because you can't leave the set (hence the gate is closed not letting anything out).

Similarly, taking any two polynomials and multiplying them together will always lead to some other polynomial. Part A shows this in action because (2x+7), (5x+9) and 10x^2 + 53x + 63 are all polynomials.

Answer 2

$\rm 10x^{2} + 53x + 63$ is the expression that represents the area of the rectangle. This is a trinomial expression.

What is a quadratic equation?

A quadratic equation is the polynomial of second degree in one variable only. The standard form of a quadratic equation is $\rm ax^{2} +bx+c=0$

Where, x is the variable and a,b, and c are real numbers.

Part A

Let length of a rectangle = (2x+7)

Let width of a rectangle = (5x+9)

The area of the rectangle = length × width

=   (2x+7)(5x+9)

= 2x × 5x + 2x × 9 + 7 × 5x + 7 × 9

= $\rm 10x^{2} + 18x + 35x + 63$

Therefore, $\rm 10x^{2} + 53x + 63$ is the expression that represents the area of the rectangle.

Part B

The degree is 2 in the expression. This is a trinomial expression.

The degree is simply the largest exponent. The degree directly represents the type of polynomial. Since the degree is 2. The polynomial is a trinomial because it has 3 terms.

Part C

Applying multiplication operations such that taking any two polynomials and multiplying them together will always lead to some other polynomial. Part A shows this multiplication because (2x+7), (5x+9) and $\rm 10x^{2} + 53x + 63$ are all polynomials.

Learn more about quadratic equations;

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