Exercises: Algebraic expressions

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

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

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

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 $y = -3$

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) $x + \frac{1}{2}x + 3x$

An architect is designing a rectangular room. The length of the room is $5$ metres 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.

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).

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

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?

Cian is working on an assignment about polynomials and needs to simplify the following expression by combining like terms:

$$P(x) = 4x^3 - 7x^2 + 2x^3 + 5$$

In his notebook, he writes down the following procedure:

$$P(x) = (4 - 7 + 2)x^{3+2+3} + 5 = -1x^8 + 5 = -x^8 + 5$$

and concludes by stating that the final result is a binomial of degree $8$.

1. a) Identify and explain in detail the two conceptual errors made by Cian in his solution.
2. b) Correct the exercise in your notebook by performing the steps correctly. What is the resulting simplified polynomial, its degree, and its constant term?

Design and write down in each case an algebraic expression that strictly satisfies all the required conditions:

1. a) A monomial involving two different variables, $x$ and $y$, whose coefficient is a negative fraction and whose total degree is equal to $5$.
2. b) A polynomial of degree $3$ with the indeterminate $x$, ordered from highest to lowest degree, that has three terms, whose leading coefficient is $4$, its constant term is $-6$, and has $x = 1$ as a root (remember that a number $a$ is a root of a polynomial if its numerical value is $0$, that is, $P(a) = 0$). Justify step by step how you deduce the missing middle term.

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.

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 journeys.

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 $C(x+3)$.

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

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.

We have a cardboard box in the shape of a cuboid (rectangular prism) whose volume is given by the algebraic expression:

$$ V(x) = x^3 + 4x^2 + 3x $$

1. Factorise out the common factor in the expression for the volume.

2. Factorise 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?

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.

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

In Cork, the public transport company offers two options for bus travel during a month:

• Option A (Single ticket): each ticket costs $1.40$ €.

• Option B (Reloadable monthly pass): a fixed maintenance fee of $12.50$ € per month is paid, and each trip taken has a reduced cost of $0.60$ €.

1. a) Write the polynomial $A(x)$ representing the monthly cost of making $x$ trips using Option A, and the polynomial $B(x)$ representing the monthly cost of making $x$ trips using Option B.
2. b) Define and simplify the polynomial $D(x) = A(x) - B(x)$, which represents the financial savings obtained by choosing the monthly pass instead of single tickets.
3. c) If a student estimates that she will make exactly $20$ trips throughout the month, calculate the numerical value $D(20)$ and interpret the result obtained in the context of the problem: which option is more economical for her and why? Starting from how many trips does the monthly pass become cost-effective?

At a Met Éireann mountain weather station, the temperature $T$ (in degrees Celsius) during a winter day is modeled using the following polynomial:

$$T(h) = -0.2h^2 + 2.4h - 3$$

where $h$ represents the number of hours elapsed since 08:00 in the morning ($h = 0$ corresponds to 08:00, $h = 4$ corresponds to 12:00, etc.).

1. a) Observe the coefficients of the polynomial before calculating. Knowing that at 08:00 ($h = 0$) the temperature was $-3$ ºC, make a reasoned estimate of whether at 12:00 ($h = 4$) the temperature will have exceeded $0$ ºC or not.
2. b) Now calculate the exact numerical value $T(4)$ and compare the result with your previous estimate.
3. c) A novice observer claims that at $18:00$ ($h = 10$) the temperature will reach $15$ ºC. Without calculating the exact value yet, estimate whether this claim is plausible taking into account the sign of the quadratic term $-0.2h^2$ for increasing values of $h$. Then, calculate $T(10)$ and confirm whether the claim was correct.

An architecture firm is designing a new rectangular town square 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 $x$ metres wide and twice as long, that is, $2x$ metres. The path surrounding the garden has a constant width of 3 metres on all sides.

1. Draw a sketch of the situation.

2. Write the simplified algebraic expression representing the total area of the square (garden plus path).

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

You can do it on paper and send a photo with your answer.

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.

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.

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.

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).

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.

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?

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 $\pi$.

We want to find the numerical value of the algebraic expression:

$$E(x) = (2x - 3) \times (x^2 + 4x - 5) - (2x^2 - x) \times (x + 4)$$

for the value $x = -2$.

Two students are discussing the best strategy to compute it:

• Cian proposes Method 1: substitute $x = -2$ directly into each of the factors of the original expression and perform the arithmetic operations following the order of operations.

• Aoife proposes Method 2: first multiply the polynomials, group and combine all like terms to obtain the expression in its simplest standard form, and then substitute $x = -2$.

1. a) Apply Cian's Method 1, detailing each of the intermediate operations.
2. b) Apply Aoife's Method 2, first expanding and simplifying the algebraic expression, and then calculating the numerical value to check that it matches the previous one.
3. c) Compare both approaches, assessing the risk of error in each. If you were now asked to calculate the value of that same expression for four different values ($x = 0$, $x = 1$, $x = 2$, and $x = 3$), which of the two methods would you choose and why? Justify your answer.

Consider the family of polynomials given by the expression:

$$P(x) = 2x^3 + kx^2 - 8x - 12$$

where $k$ is an unknown real constant.

1. a) Knowing that $x = -2$ is a root of $P(x)$, find with justification the exact value of the coefficient $k$.
2. b) Once that value of $k$ is fixed, check in a reasoned manner by calculating the numerical value whether $x = 3$ and $x = 1$ are also roots of the polynomial.
3. c) A classmate claims: «Since $P(x)$ has degree $3$, we can surely find four distinct real numbers that make the polynomial zero». Explain why this statement is mathematically false, relying on the relationship between the degree of a polynomial and its maximum number of roots.
4. d) Construct a polynomial $Q(x)$ of degree $2$ that has only the values $x = -2$ and $x = 3$ as roots, and whose leading coefficient is $-5$. Present the result ordered and simplified.