Exercises: Algebraic expressions

In an online video game shop, the price of a monthly subscription is $x$ euros. 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 voucher of $5$ euros.
3. c) What a group of friends pays for a monthly subscription plus a game costing $25$ euros.
4. d) Double the price of a subscription increased by $10$ euros.

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.

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 $x = 6$

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

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

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?

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

Aoife has to calculate the numerical value of the trinomial $3x^2 - 5x + 4$ for $x = -2$. On her exam, she wrote the following working:

1. a) Identify and clearly explain the two mistakes Aoife made regarding the use of parentheses and the order of operations.
2. b) Write the correct working step by step and find the true numerical value of the expression.

In Dublin's public electric bicycle service, registering has a fixed maintenance fee of $25$ euros per year. Each standard trip under half an hour costs $0.60$ euros, but if the user exceeds that time, an additional surcharge of $1.50$ euros applies for each extra time bracket.

1. a) Write an algebraic expression that represents the total annual expenditure of a person who takes $x$ standard trips and incurs $y$ overtime surcharges.
2. b) Classify the resulting expression according to its number of terms (monomial, binomial, or trinomial) and state its degree.
3. c) Calculate the numerical value to find out how much Cian will pay in total if, throughout the year, he took $60$ standard trips and incurred $4$ surcharges.

In a comic shop, the price of a monthly subscription is $x$ euros. For each additional comic you buy, you are charged $y$ euros, but if you buy more than five, a discount of $3$ euros is applied to the total. Write the algebraic expression that represents the monthly expenditure of a customer who has bought $7$ additional comics.

Find the algebraic expression that represents the area of the following geometric figure in terms of $x$. Simplify the result by collecting like terms.

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) $x = -2, y = -2$

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.

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) $(x^2 + 3) \times (x^2 - 3)$

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) $12x^2y^3 - 18x^3y^2$

Design a trinomial in the variable $x$ that simultaneously satisfies these four conditions:

1. Its degree is $2$.

2. The coefficient of the highest-degree term is a negative integer.

3. Its constant term is $5$.

4. Its numerical value for $x = 3$ is exactly equal to $2$.

1. a) Determine the missing coefficients with justification and write the algebraic expression of your trinomial.
2. b) Is this the only possible solution, or are there other distinct trinomials that also satisfy all the conditions? Justify your answer by deducing what relationship their coefficients must satisfy.

A tech company estimates its monthly profits (in euros) using the polynomial $B(x) = 2x^3 - 40x^2 + 150$, where $x$ represents the number of contracted server packages (in tens).

1. a) Without calculating the exact value yet, estimate whether for $x = 50$ the profit will be positive or negative and what its order of magnitude will be (hundreds of thousands or millions of euros). Justify your estimate by explaining which term dominates the polynomial when the variable becomes large.
2. b) Calculate the exact numerical value of $B(50)$ by substituting and operating in order.
3. c) Check whether your estimate was correct and now calculate the numerical value for $x = 3$. Based on this second calculation, explain why for small values of the variable the highest-degree term does not always determine the sign of the result on its own.

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.

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 grey area).

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

An electric scooter rental company charges a base fee of $B$ euros for unlocking and $0.15$ euros per minute of use ($m$). In addition, if the user travels more than $5$ kilometres, a safety surcharge of $S$ euros is applied. VAT at $21\%$ 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 $1$ euro and the safety surcharge is $2$ euros, how much will a person pay who uses the scooter for $20$ minutes and travels $6$ km?

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.

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 $x = -1$.

A play area in a park is to be designed, consisting of a large circle with radius $R$. Inside this circle, a square sandpit 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. Factorise the resulting expression by taking out the common factor.

3. If the radius $R$ is $10$ metres, 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).

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 $13$.

Two students, Aoife and Niamh, are given the task of translating into algebraic language and evaluating the following instruction: "Four times the difference between the square of a number $n$ and $9$, all increased by three times that number".

• Aoife translates the sentence preserving the literal structure using parentheses: $A(n) = 4(n^2 - 9) + 3n$.

• Niamh prefers to apply the distributive property and arrange the terms from the beginning: $N(n) = 4n^2 + 3n - 36$.

1. a) Check algebraically whether both expressions are equivalent or if either of the two made an error in her formulation.
2. b) If you have to calculate the numerical value for $n = 3$, evaluate it using Aoife's form and then using Niamh's form. Justify which of the two paths is faster and carries less risk of calculation error for this specific value.
3. c) If you are asked to identify without any ambiguity the degree of the polynomial, the number of terms, and the constant term, justify which of the two forms must be used and why.

According to data from Met Éireann, the Irish Meteorological Service, on a mountain day the ambient temperature at the starting point (at $0$ meters altitude) is $12$ ºC. As you ascend an altitude of $h$ meters, the temperature decreases by $0.65$ ºC for every $100$ meters climbed due to the vertical lapse rate. Furthermore, the wind chill feels an additional $0.2$ ºC lower for each km/h of wind speed ($v$).

1. a) Write a simplified algebraic expression that determines the wind chill $S(h, v)$ as a function of the altitude $h$ (in meters) and the wind speed $v$ (in km/h).
2. b) Classify the resulting expression according to its number of terms and state the degree of each term with respect to its variables.
3. c) A rescue team is at a mountain shelter at an altitude of $1600$ meters, and the anemometer reads wind gusts of $35$ km/h. Calculate the numerical value of the exact wind chill at the shelter.
4. d) If the Mountain Rescue protocol requires triggering an extreme cold warning when the wind chill drops below $-7$ ºC, explain algebraically above what wind speed this warning will be triggered if the team remains at that same altitude of $1600$ meters.