Algebraic Expressions

In a mobile app store, the cost of a monthly subscription to a music service is $x$ pesos. In addition, there is a fixed fee of $1.50$ 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 $2$ pesos on the base price $x$.

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.

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

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

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 cents based on the minutes of use ($x$):

$$ C(x) = 2x^2 - 5x + 150 $$

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

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

2. If a user has paid exactly 150 cents, 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 $C(x+3)$.

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) $(-4x^4 + 2x^2) + (4x^4 - x^2 + x)$

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): $x^2 + 10x + 25$

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).

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?

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.

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.

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?

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.

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

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.

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.

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.

An architecture firm is designing a new rectangular plaza 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 plaza (garden plus path).

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

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 flat 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 standard algebraic identities to simplify the expression for the area. The final result should be a very simple monomial or polynomial multiplied by $\pi$.