Algebraic expressions

In an online video game store, the price of a monthly subscription is $x$ pesos. Translate the following situations into algebraic expressions:

1. a) The cost of three monthly subscriptions.
2. b) The price of a subscription after applying a discount coupon of $300$ pesos.
3. c) What a group of friends pays for a monthly subscription plus a game costing $1\,500$ pesos.
4. d) Double the price of a subscription increased by $600$ pesos.

We are designing a new tablet whose screen is rectangular. The base measures $x$ centimeters and the height is $5$ centimeters less than the base. Write an algebraic expression for the perimeter of the screen and simplify it.

A municipal sports center has a rectangular basketball court. If the length of the court is $y$ meters and its width is half the length, write the polynomial that represents the area of the court in terms of $y$.

In a comic book store, the price of a monthly subscription is $x$ pesos. For each additional comic book you buy, you are charged $y$ pesos, but if you buy more than five, a discount of $180$ pesos is applied to the total. Write the algebraic expression that represents the monthly expenditure of a customer who has bought $7$ additional comic books.

An electric scooter rental company charges a base fee of $B$ pesos for unlocking and $9$ pesos per minute of use ($m$). In addition, if the user travels more than $5$ kilometers, a safety surcharge of $S$ pesos is applied. A tax of $21\%$ must be applied to the sum of these items.

1. Write the algebraic expression for the total cost $C$ of a trip if the user exceeds a distance of $5$ km.

2. Simplify the previous expression by performing any possible operations.

3. If the base fee is $60$ pesos and the safety surcharge is $120$ pesos, how much will a person pay who uses the scooter for $20$ minutes and travels $6$ km?

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

1. a) $5x - 3$ for $x = 4$
2. b) $x^{2} + 2x + 1$ for $x = -2$
3. c) $2(a + b)$ for $a = 3$ and $b = 5$
4. d) $\frac{10 - x}{2}$ for $x = 6$

Find the numerical value of the polynomial $P(x, y) = 3x^2y - 5xy + 2$ for the given values of the variables:

1. a) $x = 2, y = -1$
2. b) $x = 0, y = 10$
3. c) $x = -2, y = -2$

Consider the polynomial $P(x) = 2x^3 - kx^2 + 4x - 12$.

1. Determine the value of the coefficient $k$ knowing that the numerical value of the polynomial for $x = 2$ is equal to zero ($x = 2$ is a root of the polynomial).

2. Once $k$ has been found, calculate the numerical value of $P(x)$ for $x = -1$.

Observe the following sequence of patterns made with dots and determine the algebraic expression for the general term $a_{n}$ (where $n$ is the position of the pattern). How many dots will the pattern in position $n=50$ have?

A digital artist is creating a pixel mosaic that grows following a certain pattern. In stage $n=1$ they use 3 pixels, in stage $n=2$ they use 8 pixels, in stage $n=3$ they use 15 pixels, and so on.

Look at the visual pattern and complete the following tasks:

1. Find the algebraic expression (nth term) that determines the number of pixels required for any stage $n$.

2. If the digital canvas has a limit of $1000$ pixels, could the artist complete stage $n=30$? Justify your answer by calculating the numerical value.

Identify the parts of the following monomials by completing the information for each:

1. a) $7x^{2}$
2. b) $-5ab^{3}$
3. c) $x^{4}y$
4. d) $\frac{3}{4}z^{2}$

For each one, state the Coefficient, Literal part and Degree.

Classify the following algebraic expressions, indicating whether they are monomials, binomials or trinomials. In the case of polynomials (binomials and trinomials), also state their degree.

1. a) $4x^{3} - 5x + 2$
2. b) $-7ab^{2}$
3. c) $x^{2} + 10$
4. d) $3y^{4} - 2y^{2}$

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

1. a) $3x + 5x - 2x$
2. b) $7a^{2} - 3a + 2a^{2} + 5a$
3. c) $4xy + 2x - xy + 3x$
4. d) $10n - 5 - 3n + 8$

Simplify the following algebraic expression as far as possible by operating with monomials and removing the brackets:

$3x \times (2x^2) - 4x^2 \times (x - 5) + \frac{12x^4}{3x}$

Given the polynomials $A(x) = 4x^3 - 6x^2 + 5x$ and $B(x) = -2x^3 + 8x^2 - 7$, perform the following operations and write the resulting polynomial in descending order of degree:

1. a) $A(x) + B(x)$
2. b) $A(x) - B(x)$
3. c) $2 \times A(x) + 3 \times B(x)$

A surveillance drone flies over a rectangular plot whose dimensions are given by the polynomials $L(x) = x^2 - 3x + 5$ for the length and $A(x) = 2x + 1$ for the width. Obtain the polynomial that represents the total area of the plot.

The new model of a smartphone has a rectangular glass screen with dimensions of $x$ in width and $x+4$ in length. Around the screen there is a protective frame or bezel with a uniform thickness of $2$ cm on all sides.

1. Find the simplified polynomial that expresses the total area of the front of the phone (screen plus bezel).

2. Find the simplified polynomial that expresses only the area of the bezel (the gray area).

3. If the width of the screen $x$ is $6$ cm, what is the area of the bezel?

Use algebraic identities to expand the following expressions without multiplying step by step:

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

Expand and simplify the following algebraic expression as far as possible using algebraic identities to collect like terms:

Show the process step by step, indicating which algebraic identity you have used in each case.

Prove algebraically that the difference between the squares of two consecutive integers is always equal to the sum of those numbers.

Instructions:

1. Call the first number $n$ and the next number $n+1$.

2. Write the expression for the difference of their squares: $(n+1)^2 - n^2$.

3. Expand and simplify the expression.

4. Check the result with the numbers $12$ and $13$.

Factorise the following algebraic expressions as far as possible by taking out the highest common factor:

1. a) $15x^4 - 10x^3 + 5x^2$
2. b) $6a^2b - 9ab^2 + 3ab$
3. c) $12x^2y^3 - 18x^3y^2$

A play area in a park is to be designed, consisting of a large circle with radius $R$. Inside this circle, a square sandbox will be installed whose side length is exactly the radius $R$ of the circle. The rest of the circle will be covered with artificial grass.

1. Write the simplified algebraic expression for the area of the artificial grass zone in terms of $R$. Use $\pi$ as a symbol.

2. Factorize the resulting expression by taking out the common factor.

3. If the radius $R$ is $10$ meters, calculate the grass area (use $\pi \approx 3.14$).

Simplify the following algebraic fraction by first taking out common factors and using algebraic identities in the numerator and the denominator:

For what value of $x$ would this expression be undefined? (Hint: the denominator cannot be zero).