Exercises: Algebraic expressions

Identify the coefficient, the literal part and the degree of the following monomial: $M(x,y) = -\frac{2}{3}x^4y^2$.

Calculate the numerical value of the polynomial $P(x) = 3x^2 - 5x + 2$ for $x = -2$.

Express the following statement in algebraic terms: 'The cube of a number minus the square of another, different number'.

State which of the following algebraic expressions are polynomials and justify your answer:

• $A(x) = 3x^2 - \frac{5}{x}$

• $B(x) = \sqrt{2}x^3 - x$

• $C(x) = 4x^{1/2} + 1$

Perform the following addition and subtraction of like terms and simplify the result: $3x^2y - 5x^2y + 8x^2y$.

Multiply the following monomials: $(-3a^2b^3) \cdot (4ab^4)$.

Determine the value of $k$ such that the numerical value of the polynomial $P(x) = 2x^3 - kx + 4$ at $x = 1$ is equal to $5$.

Write the algebraic expression that represents the perimeter of a rectangle whose base is $3x$ and whose height is $x + 2$.

Marcos has to simplify the algebraic expression $E(x, y) = 4x^3y^2 + 2x^3y^2 - 3x \cdot (2x^2y^2)$ in an exam. On his paper, he writes the following working:

Step 1: $4x^3y^2 + 2x^3y^2 = 6x^6y^4$.

Step 2: $3x \cdot (2x^2y^2) = 6x^2y^2$.

Step 3: $6x^6y^4 - 6x^2y^2 = 0x^4y^2 = 0$.

1. a) Identify and explain with reasoning the conceptual errors Marcos makes in each of the three steps, indicating which properties of operations with monomials he has violated.
2. b) Carry out the correct simplification step by step, state whether the obtained result is a monomial or a polynomial, and determine its absolute degree.

The aim is to determine the formula of a two-variable monomial of the form $M(x, y) = c \cdot x^{a+2} y^{3a-1}$, where $c$ is a real number and $a$ is a natural number.

1. a) Knowing that the absolute degree of the monomial is $9$, set up a linear equation in the unknown $a$ and find its value.
2. b) Using the value of $a$ found, calculate the coefficient $c$ knowing that the numerical value of the monomial for $x = -1$ and $y = 2$ is equal to $-64$.
3. c) If you keep the value of $y = 2$ but triple the value of $x$ making it $x = -3$, explain by what factor the final result is multiplied without needing to recalculate all the powers from scratch.

A clothes shop offers a discount of $20\%$ on all its items. If you buy $x$ shirts that originally cost $25$ euros each and $y$ pairs of trousers that cost $40$ euros each, write the algebraic expression for the total cost of the purchase. Calculate the cost if you buy $3$ shirts and $2$ pairs of trousers.

The kinetic energy of a moving body is given by the expression $E_c = \frac{1}{2}mv^2$, where $m$ is the mass in kilograms and $v$ the speed in metres per second. Calculate the kinetic energy of a $800$ kg vehicle travelling at $20$ m/s.

The figure shows a square with side length $2x$ containing inside it a circle of radius $x$. Write the algebraic expression that represents the area of the shaded region (the region of the square outside the circle).

Calculate the numerical value of the algebraic expression $P(x,y) = \frac{x^2 - 2xy + y^2}{x - y}$ for $x = \frac{1}{2}$ and $y = -\frac{3}{2}$. Hint: simplify the expression before substituting the values.

A phone company charges a fixed monthly fee of $15$ euros, plus $0,12$ euros for each minute of calls $m$ and $0,05$ euros for each megabyte consumed $d$. Write the algebraic expression for the monthly bill and calculate its total for a month in which $100$ minutes and $500$ megabytes have been used.

Find the values of the parameters $a$ and $b$ in the polynomial $P(x) = ax^2 + bx - 5$ knowing that its numerical value for $x = 1$ is $-2$ and for $x = -1$ is $-4$.

A cylindrical tank has a base radius $r = x$ and a height $h = 3x$. Write the algebraic expressions representing its volume and its curved surface area in terms of $x$.

Perform the following division of monomials and state the coefficient, literal part and degree of the resulting monomial: $\frac{-24x^5y^4z}{6x^2y^3}$.

A sustainable bicycle courier cooperative in Sevilla offers small businesses a monthly rate consisting of a fixed fee of $35$ euros, plus $2,40$ euros for each standard delivery (under $3$ kg) and $4,50$ euros for each urgent or bulky delivery. In addition, to encourage local trade, if more than $80$ deliveries in total are made in a month, a $15\%$ discount is applied exclusively to the delivery costs (the fixed fee is not discounted).

1. a) Define the two variables involved and write the polynomial that expresses the monthly cost $C_1(x, y)$ of a business that does not exceed $80$ total deliveries. Identify the constant term and explain its practical meaning.
2. b) Write the simplified polynomial $C_2(x, y)$ that represents the monthly cost if the business exceeds $80$ deliveries in the month.
3. c) A downtown bookstore made $50$ standard deliveries and $40$ urgent deliveries last month. Justify which formula you must apply, calculate the total invoice amount, and interpret what percentage of the total expenditure corresponded to the fixed fee.

In an environmental control centre in Castilla y León, the concentration of a pollutant in the air (in $\mu\text{g}/\text{m}^3$) during an $8$-hour workday is modelled by the polynomial $C(t) = -0,25 t^3 + 2,4 t^2 - 3,5 t + 18$, where $t$ represents the time in hours elapsed since 08:00 ($0 \leq t \leq 8$). We want to find the concentration at 12:00 ($t = 4$).

1. a) Before working with exact values, make an estimate by rounding the coefficients with decimals to the nearest integer (in case of a tie, as with $-3,5$, round to $-4$) and calculate mentally whether the value of $C(4)$ will be higher or lower than the initial concentration $C(0) = 18\,\mu\text{g}/\text{m}^3$. Justify your prediction.
2. b) Calculate the exact numerical value of $C(4)$, following the order of operations and the rules of signs.
3. c) Determine the absolute error made in your estimate ($|\text{exact value} - \text{estimated value}|$) and explain, considering the signs of the monomials in the polynomial, which terms drove the increase of the pollutant and which one slowed down its growth.

An open-topped box is to be made from a rectangular cardboard sheet measuring $40$ cm long by $30$ cm wide. To do this, squares with side length $x$ are cut out from each of the four corners and the flaps are folded upwards. Write the algebraic expression for the volume of the box in terms of $x$ and calculate this volume if $x = 5$ cm.

Prove algebraically that the difference between the squares of two consecutive odd numbers is always a multiple of $8$. (Remember that an odd number can be written in the form $2n+1$).

Given the algebraic expression $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, simplify it fully by carrying out the division and use the result to calculate its numerical value mentally when $a = 4,5$ and $b = 1,5$.

In an electrical circuit, the equivalent resistance $R_{eq}$ of three resistors $R_1$, $R_2$ and $R_3$ connected in parallel is obtained from the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$. Rearrange for $R_{eq}$ and write its algebraic expression as a single fraction in terms of $R_1$, $R_2$ and $R_3$.

A rectangular garden has dimensions of $a$ metres long by $b$ metres wide. Two perpendicular paths of width $x$ are built parallel to the sides of the garden, forming a cross. Write the factorised algebraic expression for the area of the remaining green area.

Two car rental agencies offer different rates. Agency A charges a fixed fee of $30$ euros plus $0,20$ euros per kilometre travelled. Agency B charges a fixed $50$ euros plus $0,15$ euros per kilometre. Write the algebraic expression representing the difference in cost between Agency A and Agency B for a journey of $k$ kilometres, and determine for how many kilometres the cost is exactly the same.

Write the algebraic expression that gives the sum of the first $n$ positive integer multiples of $3$, knowing that the sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$. Use this expression to calculate the sum of the first $50$ multiples of $3$.

A farmer has $L$ metres of fencing to enclose a rectangular plot of land adjacent to a straight river. Since the river acts as a natural barrier, no fencing is needed along that side. If we call $x$ the width of the plot (the dimension perpendicular to the river), write the algebraic expression for the area of the enclosure in terms of $L$ and $x$. What is the area if he has $100$ metres of fencing and the width is $20$ metres?

We want to calculate the numerical value of the rational algebraic expression $E(x, y) = \dfrac{20x^5y^3 - 8x^3y^5 + 16x^4y^4}{4x^2y^2}$ for the values $x = 2,5$ and $y = -2$. To solve it, two students propose different methods:

• Sergio's method: substitute the values $x = 2,5$ and $y = -2$ directly into the numerator and denominator, calculate the powers with exponents $5$, $4$, and $3$, carry out the multiplications and additions, and finally divide the resulting numerator by the denominator.

• Alba's method: previously divide each term of the numerator by the monomial in the denominator applying the rules for dividing monomials to obtain a simplified polynomial, and substitute the numerical values only into the resulting expression.

1. a) Fully carry out Alba's method until you obtain the simplified polynomial, and calculate its numerical value for the indicated values.
2. b) Analyse the number of operations with high-exponent powers and decimal multiplications required by Sergio's method compared to Alba's. Critically justify which of the two methods is more convenient in a test without a calculator to minimise time and the risk of error.
3. c) Would Alba's method be valid if the assigned values were $x = 0$ or $y = 0$? Justify your answer based on the definition of the original expression.

Let $P(x)$ and $Q(x)$ be two non-zero single-variable polynomials, with respective degrees $\operatorname{gr}(P) = m$ and $\operatorname{gr}(Q) = n$, such that $m > n \geq 1$.

1. a) The polynomials $A(x) = [P(x)]^2 \cdot Q(x)$ and $B(x) = P(x) \cdot [Q(x)]^2$ are formed. Knowing that the degree of $A(x)$ is $11$ and the degree of $B(x)$ is $10$, set up a system of two linear equations in the unknowns $m$ and $n$ using the properties of the degree of a product and power of polynomials, and deduce the original degrees of $P(x)$ and $Q(x)$.
2. b) Consider the polynomial $C(x) = P(x) - k \cdot x^{m-n} \cdot Q(x)$, where $k$ is a non-zero real constant. Can it always be guaranteed that the degree of $C(x)$ is equal to $m$? Theoretically justify your answer and provide a specific numerical counterexample with simple polynomials where $\operatorname{gr}(C) < m$, stating what precise mathematical condition $k$ must satisfy with respect to the leading coefficients of $P(x)$ and $Q(x)$ for such a reduction of degree to occur.