Monomials

To understand how mathematical expressions work in economics, it is essential to identify their components. Complete the requested information for the following monomials:

1. a) $7x^3$

1. b) $-x^2y$

1. c) $\frac{3}{5}ab^4$

For each of them, state the coefficient, the literal part and the degree.

An architect is designing a rectangular window for a bank office. The dimensions of the window are given as a function of a variable $x$ as shown in the image. Write the monomial that represents the area of the window.

In user interface design, variables are sometimes used to define the size of elements. In order to add or subtract two elements, their monomials must be like terms.

Determine which of the following pairs of monomials are like terms:

1. a) $5x^2y$ and $-2x^2y$

1. b) $3ab^2$ and $3a^2b$

1. c) $-x$ and $\frac{1}{2}x$

1. d) $4z^3$ and $4z^2$

Simplify the following expressions by operating with monomials and reduce the result to its simplest form:

1. a) $\displaystyle \frac{(15x^3y^2) \cdot (-2xy^4)}{6x^2y^3}$

1. b) $\displaystyle -3a^2b \cdot \left(\frac{1}{3}ab^3\right) + \frac{10a^3b^4}{2}$

In a packaging design company, cardboard boxes in the shape of a rectangular prism are being designed. The length of the base is three times the width ($x$) and the height of the box is half the width.

1. Express the perimetre of the base of the box as a monomial in terms of the variable $x$.

2. Express the total volume of the box as a monomial. State the coefficient, the literal part and the degree of the monomial obtained.

In a rectangular plot intended for urban allotments, an automatic irrigation system is to be installed. The main plot has dimensions $5x$ long by $3y$ wide. Inside it, a square space with side length $2x$ is reserved for a tool shed that does not require watering.

1. Find the monomial that represents the total area of the plot.

2. Find the polynomial that represents the area to be covered by the irrigation system.

3. Calculate the irrigation area if $x = 4$ meters and $y = 6$ meters.

Simplify the following expressions by collecting like terms, just as you would when calculating the total balance of income and expenses for a small shop:

1. a) $3x^2 + 5x^2 - 2x^2$

1. b) $7ab - 3ab + ab$

1. c) $4x^3 - 5x + 2x^3 + 9x$

Determine whether the following statements are true or false, justifying your answer with examples or counterexamples:

1. a) The sum of two monomials of degree 3 is always another monomial of degree 3.
2. b) The degree of the monomial resulting from multiplying a monomial of degree $n$ by another of degree $m$ is $n + m$.
3. c) When dividing two monomials with the same literal part, the result is always a real number (degree 0).

A financial analyst models the profits of a tech startup using the monomial $B(x, y) = 0,45x^2y$, where $x$ represents the number of active users (in thousands) and $y$ the average marketing spend per user (in euros).

1. Calculate the numerical value of the profit if the company has $20.000$ users ($x=20$) and the marketing spend is $10$ € per user ($y=10$).

2. If the number of users doubles and the marketing spend is halved, what happens to the total profit? Justify your answer using the properties of monomials.

In an artificial intelligence data centre, the volume of air needed to cool a server rack is modelled by the product of three critical dimensions. The height is a monomial $A(x, y) = \frac{3}{4}x^2y$, the width is $B(x, y) = \frac{8}{9}xy^3$ and the depth is $C(x, y) = \frac{1}{2}x^3y^2$.

Obtain the resulting monomial that defines the total volume $V(x, y)$ of the rack and determine its coefficient, its literal part and its overall degree. What would happen to the degree of the monomial if we doubled the exponent of the variable $x$ in the depth?