Exercises: Algebraic expressions

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

1. a) The cost of three monthly subscriptions.
2. b) The price of a subscription after applying a discount coupon of $\$5$.
3. c) What a group of friends pays for a monthly subscription plus a game of $\$25$.
4. d) Double the price of a subscription increased by $\$10$.

Identify the elements of the following monomials by completing the information for each one:

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

For each one, state: Coefficient, Variable part and Degree.

Evaluate the following algebraic expressions for the indicated 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 $x = 6$

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

Reduce 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) $10n - 5 - 3n + 8$

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

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) $3y^{4} - 2y^{2}$

A community sports center has a rectangular basketball court. If the length of the court is $y$ feet and its width is half the length, write the polynomial that represents the area of the court as a function of $y$.

In a comic book shop, the price of a monthly subscription is $\$x$. For each additional comic you buy, they charge you $\$y$, but if you buy more than five, they apply a discount of $\$3$ to the total. Write the algebraic expression that represents the monthly cost of a customer who has bought $7$ additional comics.

Calculate the algebraic expression that represents the area of the following geometric figure as a function of $x$. Simplify the result by combining like terms.

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

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

Reduce the following algebraic expression as much as possible by operating with monomials and removing the parentheses:

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

Given the polynomials $A(x) = 4x^3 - 6x^2 + 5x$ and $B(x) = -2x^3 + 8x^2 - 7$, perform the following operations and order the resulting polynomial from highest to lowest degree:

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

A surveillance drone flies over a rectangular lot 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 lot.

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

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

Factor out the greatest common factor in the following algebraic expressions:

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

A digital artist is creating a pixel mosaic that grows following a specific 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.

Observe the visual pattern and complete the following tasks:

1. Find the algebraic expression (general term) that determines the number of pixels needed 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.

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

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

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

3. If the width of the screen $x$ is $6$ in., what is the area of the frame?

An electric scooter rental company charges a base fare of $\$B$ for unlocking and $\$0,15$ for each minute of use ($m$). Furthermore, if the user travels more than $5$ miles, a safety surcharge of $\$S$ is applied. A $21\%$ sales tax must be applied to the sum of these amounts.

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

2. Simplify the previous expression by carrying out the possible operations.

3. If the base fare is $\$1$ and the safety surcharge is $\$2$, how much will a person pay who uses the scooter for $20$ minutes and travels $6$ miles?

Operate and simplify the following algebraic expression as much as possible using algebraic identities to reduce terms:

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

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 $k$ is found, calculate the numerical value of $P(x)$ for $x = -1$.

A play area in a park is to be designed consisting of a large circle of radius $R$. Inside this circle, a square sandbox will be installed whose side measures 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 as a function of $R$. Use $\pi$ as a symbol.

2. Factor out the common factor in the expression obtained.

3. If the radius $R$ is $10$ feet, calculate the grass area (use $\pi \approx 3,14$).

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

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

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

Instructions:

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

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 $13$.