Exercises: 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 $x$.

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

Monomial	Coefficient	Variable Part	Degree
$-5x^2y$
$\frac{3}{4}ab^3$
$x^2yz^2$
$-12$

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 $y = -3$

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) $x + \frac{1}{2}x + 3x$

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.

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).

Perform the following additions and subtractions of polynomials:

1. a) $(3x^2 - 5x + 4) + (2x^2 + x - 6)$
2. b) $(x^3 - 2x^2 + 3) - (x^2 - 5x + 1)$
3. c) $(-4x^4 + 2x^2) + (4x^4 - x^2 + x)$

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?

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.

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?

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.

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

$$ C(x) = 40x^2 - 100x + 3,000 $$

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 $C(x+3)$.

Consider the following polynomials representing the dimensions and properties of different geometric solids:

$$ P(x) = 3x^2 - 5x + 1 $$ $$ Q(x) = 2x^2 + 4x - 2 $$ $$ R(x) = x - 3 $$

Perform the following combined operations and simplify the final result:

1. Calculate the difference: $P(x) - Q(x)$.

2. Calculate the product: $Q(x) \times R(x)$.

3. Calculate: $2 \times P(x) - 3 \times Q(x)$.

Using the notable identities (square of a sum, square of a difference and sum times difference), expand or factor the following algebraic expressions as appropriate. Relate part 2 to the geometric interpretation shown in the image.

1. Expand: $(3x + 2)^2$

2. Expand: $(x - 4)^2$

3. Expand: $(2x + 5)(2x - 5)$

4. Factor the polynomial (write it as the square of a binomial): $x^2 + 10x + 25$

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.

We have a cardboard box in the shape of a rectangular prism whose volume is given by the algebraic expression:

$$ V(x) = x^3 + 4x^2 + 3x $$

1. Factor out the common factor in the expression for the volume.

2. Factor completely the resulting polynomial to find the algebraic expressions corresponding to the length, width and height of the box.

3. If we know that $x = 5$ cm, what are the actual numerical dimensions of the box?

Find the value of the constant $k$ in the following cases, applying the Remainder Theorem:

1. So that the division of the polynomial $P(x) = x^3 - 3x^2 + kx - 2$ by $(x+2)$ is exact (that is, the remainder is 0).

2. So that the remainder of dividing $Q(x) = 2x^2 + kx + 5$ by $(x-1)$ is equal to 4.

Simplifying algebraic fractions. Remember that to simplify, you must first factorise both the numerator and the denominator using standard identities or by taking out a common factor.

Simplify the following algebraic fractions:

1. $$ \frac{4x^2 - 9}{2x + 3} $$
2. $$ \frac{x^2 - 4x + 4}{x^2 - 4} $$
3. $$ \frac{x^3 + 2x^2}{x^2 + 4x + 4} $$

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.

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.

An architecture firm is designing a new rectangular town square for a school. The design consists of a central rectangular garden area surrounded by a paved path of uniform width.

We know that the inner garden area has dimensions of $x$ meters wide and twice as long, that is, $2x$ meters. The path surrounding the garden has a constant width of 3 meters on all sides.

1. Draw a sketch of the situation.

2. Write the simplified algebraic expression representing the total area of the square (garden plus path).

3. Write the algebraic expression representing the area of the paved path (the shaded area in the drawing).

We have a polynomial $P(x) = 2x^4 - 5x^3 + kx^2 - 12$. We know that when dividing this polynomial by the binomial $(x + 2)$, the remainder obtained is 40.

Using the Remainder Theorem (without performing long division or full Ruffini's rule), determine the value of $k$. Once the value of $k$ is found, check whether $x=2$ is a root of the resulting polynomial.

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.

Look at the following sequence of figures formed by dots.

• Figure 1: Has 3 dots (forming a triangle).

• Figure 2: Has 6 dots.

• Figure 3: Has 10 dots.

• Draw Figure 4 following the growth pattern.

• Find an algebraic expression (polynomial) in terms of $n$ that allows us to calculate the total number of dots in Figure n. Hint: Relate these numbers to the area of a rectangle with sides $n+1$ and $n+2$.

• Use your expression to calculate how many dots Figure 20 would have.

You can do it on paper and send a photo with your answer.

Analyse the following incomplete standard algebraic identities. Your task is to find the value of the missing terms (represented by gaps) and write the complete identity.

1. $( \dots + 3y)^2 = 4x^2 + \dots + 9y^2$

2. $(5x - \dots)^2 = \dots - 20x + 4$

3. $(3x + \dots) \times (3x - \dots) = 9x^2 - 16$

Briefly explain which formula you used in each case (square of a sum, square of a difference, or difference of two squares).

The area of a rectangle is given by the polynomial $A(x) = 2x^2 + 7x + 3$.

1. Factorise the polynomial $A(x)$ to find the algebraic expressions corresponding to the base and height of the rectangle, knowing that both expressions are binomials with integer coefficients.

2. If we know that the value of $x$ is such that $x > 0$, calculate what the perimeter of the rectangle would be in terms of $x$. Simplify the resulting expression as much as possible.

We want to prove algebraically a mathematical curiosity about odd numbers.

We know that any odd number can be written in the form $2n + 1$ (where $n$ is an integer) and its consecutive odd number would be $2n + 3$.

1. Write the algebraic expression for the difference of the squares of two consecutive odd numbers: $(2n+3)^2 - (2n+1)^2$.

2. Expand and simplify this expression as much as possible.

3. Based on the simplified result, reason: Is it true that the difference of the squares of two consecutive odd numbers is always a multiple of 8?

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 $\pi$.

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?

A workshop in Guadalajara makes open-top boxes for tamarind candy from square sheets of cardboard $30$ cm on a side: they cut a square of side $x$ cm out of each corner and fold the flaps up.

1. a) Write the side of the bottom of the box and its height in terms of $x$.
2. b) Write the volume of the box as a polynomial in $x$, in descending order of degree.
3. c) Which box has the greater volume, the one with $x = 3$ cm or the one with $x = 5$ cm?
4. d) Explain why $x$ must be less than $15$.