Algebraic expressions

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.

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

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.

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.

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.

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?

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 $x = 6$

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.

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.

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

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) $2 \times A(x) + 3 \times B(x)$

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

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

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

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 $13$.

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).