Question

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 to receive full credit.

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

@phi

Answer 2

Well, the area of a rectangle is simply the product of the two sides, in this case,
(4x+5)(3x+10) sq.units.
Expand the product to get the actual expression.

It will be a degree 2 polynomial.
Assuming, we are talking about the ring of polynomials with real coefficients, see that the product is also a polynomial with real coefficients so we have closure under multiplication ( can say the same if we are talking about the ring of polynomials with integer or rational coefficients)

Answer 3

ok so could we start with each part so may we first start with part a?

Answer 4

Sure, for part a, expand the product (4x+5)(3x+10) and get the expression.

Answer 5

ok could you show me?

Answer 6

Well, \((4x+5) (3x+10) = 12x^2 + 40x + 15x + 50\) (using distributive law of multiplication)
So, collecting similar terms, the final expression is \(12x^2 + 55x +50\)

Answer 7

So the area will be \((12x^2 + 55x +50)\) square units.

Answer 8

may you possibly show me the work it helps me better?

Answer 9

Well, multiplying the two expressions goes like this:
(4x+5)(3x+10) = 4x*(3x+10) + 5*(3x + 10)
= 4x*3x + 4x*10 + 5*3x + 5*10
= \(12x^2 + 40x + 15x + 50 \)
= \( 12x^2 + 55x + 50\)
Hope, that clears it.

Answer 10

srry i left :( may you help with part b? now ?

Answer 11

@adxpoi

Answer 12

Sure, so what's the degree of the polynomial expression we obtaine in part a?