Algebraic Expressions

In a cell phone app store, the cost of a monthly subscription to a music service is $x$ pesos. In addition, there is a fixed fee of $30$ pesos for management expenses on each bill (one bill is issued per person per month). Express in algebraic language:

1. a) The total cost of one month's subscription.
2. b) The cost of three months' subscription for the same user.
3. c) The cost of one month's subscription for a group of 4 friends if each of them receives a discount of $40$ pesos on the base price $x$.

An architect is designing a rectangular room. The length of the room is $5$ meters more than the width. If we let $x$ be the width of the room:

1. a) Write an algebraic expression representing the perimeter of the room.
2. b) Write an algebraic expression representing the area of the room.

Copy and complete the following table, identifying the components of each given monomial:

Simplify the following algebraic expressions by adding or subtracting like terms:

1. a) $3x^2 - 5x + 2x^2 + x - 4x^2$
2. b) $2ab^2 + 3a^2b - 5ab^2 + a^2b$
3. c) $x + \frac{1}{2}x + 3x$

Given the polynomial $P(x) = -2x^3 + 5x^2 - x + 7$, answer the following questions:

1. a) What is the degree of the polynomial?
2. b) What is the coefficient of the term of degree 2?
3. c) What is the constant term?
4. d) Order the polynomial in ascending order (from lowest to highest degree).

Calculate the numerical value of the following algebraic expressions for the given values of the variables:

1. a) $3x^2 - 5x + 2$ for $x = 2$
2. b) $2(a + b)^2$ for $a = -1$ and $b = 3$
3. c) $\frac{x + 2y}{2}$ for $x = 4$ and $y = -3$

An electric scooter rental company uses the following algebraic expression to calculate the rental cost in centavos based on the minutes of use ($x$):

Note: This formula is a promotional model for VIP users and is valid for short journeys.

1. Calculate the numerical value of the cost if you use the scooter for 10 minutes.

2. If a user has paid exactly 3,000 centavos, how many minutes in theory has he been using the scooter according to the formula? (Hint: look at the constant term and find all possible solutions).

3. Expand the cost expression for a time of $(x + 3)$ minutes, that is, calculate and simplify $C(x+3)$.

We want to check whether certain values are "roots" of a polynomial. Remember that a root is a value that makes the polynomial equal to zero.

Given the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$:

1. Use Ruffini's rule to divide $P(x)$ by $(x - 1)$. What is the remainder? Is $x=1$ a root of the polynomial?

2. Without using Ruffini, calculate the numerical value $P(-2)$. Is $x=-2$ a root?

3. Write the polynomial $P(x)$ as the product of its irreducible factors given its roots.

A rectangular lot in Querétaro is $x$ meters wide, and its length is $4$ meters more than its width.

1. a) Write the area of the lot as a polynomial in $x$.
2. b) Find the area of the lot if the width is $x = 6$ m.

A souvenir shop in Teotihuacan sells obsidian pyramids with a square base of side $x$ cm and a height that is twice the side of the base.

1. a) Write the area of the base and the height of the pyramid in terms of $x$.
2. b) Use $V = \dfrac{1}{3} A_{\text{base}} \times h$ to write the volume of the pyramid as a monomial in $x$.
3. c) Find the volume of a pyramid with $x = 6$ cm.

Look at the following figure formed by a square of side length $x$ and an attached rectangle with base $3$ and height $x$.

1. a) Write the polynomial representing the total area of the figure.
2. b) If the side length $x$ is $4$ cm, what is the total area of the figure?

Look at the following figure composed of two joined rectangles. The lengths are given in terms of an unknown variable $x$.

Your task is to:

1. Find the simplified algebraic expression (polynomial) that represents the total perimeter of the figure.

2. Find the simplified algebraic expression (polynomial) that represents the total area of the figure.

A parcel delivery company wants to manufacture open-topped boxes from rectangular sheets of cardboard 40 cm long by 30 cm wide. To do this, four identical squares of side length $x$ are cut out from the corners and the resulting flaps are folded upwards.

1. Write the polynomial expressing the volume of the box in terms of $x$.

2. Determine the volume of the box if we cut out squares of side length 5 cm.

3. Would it make physical sense to cut out squares of side length 16 cm? Reason your answer based on the dimensions of the sheet.

A metal part in the shape of an annulus (a thin ring) is to be manufactured.

The outer radius of the ring is $R$ and the radius of the inner hole is $r$.

1. Write the algebraic expression for the area of the part ($A$) using $\pi$, $R$ and $r$.

2. Knowing that the outer radius is $x + 3$ and the inner radius is $x - 3$:

   a) Substitute these values into the formula for the area.

   b) Use special products to simplify the expression for the area. The final result should be a very simple monomial or polynomial multiplied by $\pi$.

The area of a triangle is $A = \dfrac{b \times h}{2}$.

1. a) Solve for the base $b$ in terms of the area $A$ and the height $h$.
2. b) A triangular sail for a boat in Acapulco has an area of $24$ m² and a height of $6$ m. How long is its base?

A seed container is shaped like a pyramid with a rectangular base of $12$ cm by $10$ cm and a capacity of $600$ cm³.

1. a) Starting from $V = \dfrac{1}{3} B \times h$, where $B$ is the area of the base, solve for the height $h$ in terms of $V$ and $B$.
2. b) Find the area of the base and the height of the container.

A factory in Puebla packs jalapeño peppers in cylindrical cans that are $10$ cm tall. The volume of a cylinder is $V = \pi r^2 h$. Use $\pi \approx 3.14$.

1. a) Solve for the radius $r$ in terms of $V$ and $h$.
2. b) If each can must hold $785$ cm³, what is the radius of the can?
3. c) The paper label covers the whole lateral surface of the can, which measures $2\pi r h$. How many cm² of paper does each label use?