Exercises: Polynomials

Calculate the numerical value of the polynomial $P(x, y) = 3x^2y - 5xy^2 + 2x - 3y + 1$ for $x = -2$ and $y = 3$.

Given the polynomials $A(x) = 4x^3 - 2x^2 + 5x - 1$ and $B(x) = -x^3 + 3x^2 - 2x + 4$, calculate the simplified expression of $2 \cdot A(x) - 3 \cdot B(x)$.

Expand the following product of polynomials and collect like terms to obtain the final polynomial: $(2x^2 - 3x + 4) \cdot (x^2 + 5x - 2)$.

Divide the polynomial $P(x) = x^4 - 3x^3 + 2x^2 - 5x + 7$ by $Q(x) = x^2 - x + 2$, clearly indicating the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$.

Use Ruffini's rule to find the quotient and remainder of the division $(2x^4 - 5x^3 + 4x - 6) : (x - 2)$.

Without performing the division, determine the remainder obtained when dividing the polynomial $P(x) = 3x^5 - 2x^3 + x^2 - 4$ by $x + 1$.

Using the factor theorem, check whether $x = -3$ is a root of the polynomial $P(x) = x^3 + 4x^2 + x - 6$.

Factorise out common terms and use algebraic identities to fully factorise the polynomial $P(x) = 5x^3 - 30x^2 + 45x$.

Carlos has to calculate the numerical value of the polynomial $P(x) = -2x^3 + 3x^2 - 5x + 4$ for $x = -2$. In his notebook, he wrote the following steps:

Locate the specific mistake Carlos made when applying the order of operations or the signs of powers, explain in your own words why his deduction is incorrect, and calculate the correct numerical value.

A delivery van rental company in Valencia sets its rate in euros using the polynomial $C(t) = 0,05t^2 + 0,4t + 15$, where $t$ represents the rental time in hours.

1. a) Identify the degree of the polynomial, its leading coefficient, and its constant term.
2. b) Explain what real economic meaning the constant term has in the context of renting the van.
3. c) Justify whether the polynomial is complete or incomplete.
4. d) Calculate how much a customer will pay if they use the van for $3$ hours.

The base of a rectangular solar panel measures $2x + 5$ metres and its height measures $3x - 2$ metres. Write the algebraic expressions (simplified polynomials) that represent the perimeter and area of the solar panel.

Determine the value of the parameter $k$ so that the polynomial $P(x) = x^3 - 2x^2 + kx + 8$ is divisible by $x - 4$.

An open-topped box is to be constructed from a rectangular cardboard sheet $40$ cm long by $30$ cm wide, cutting out a square of side length $x$ at each corner and folding up the flaps. Express the volume of the box as a polynomial in terms of $x$ and determine its degree.

Given the polynomials $P(x) = x^2 - 4$, $Q(x) = x + 2$ and $R(x) = 3x^2 - x + 1$, calculate and simplify the algebraic expression resulting from $\frac{P(x)}{Q(x)} \cdot R(x)$.

Fully factorise the polynomial $P(x) = x^3 - 4x^2 - 11x + 30$ knowing that one of its integer roots is among the positive factors of $30$.

The height $h$ (in metres) of a drone as a function of time $t$ (in seconds) is given by the polynomial $h(t) = -2t^2 + 12t + c$. If at $3$ seconds the drone is at a height of $25$ metres, calculate the value of $c$ and determine the initial height from which it took off.

Simplify the following algebraic fraction by first factorising the numerator and the denominator: $\frac{2x^2 - 8}{x^2 - 4x + 4}$.

A technology company estimates that its monthly revenue (in thousands of euros) is given by the polynomial $I(x) = -2x^2 + 50x$, and its production costs by $C(x) = 10x + 150$, where $x$ is the quantity of devices manufactured (in hundreds). Obtain the polynomial representing the monthly profit $B(x)$ and calculate the exact profit if $1000$ devices are manufactured.

According to the records of an AEMET weather station in the Sierra de Guadarrama during a winter day, the temperature $T$ (in degrees Celsius) between midnight and midday is approximated by the polynomial $T(t) = -0,1t^2 + 1,2t - 5$, where $t$ is the elapsed time in hours ($0 \le t \le 12$).

1. a) Without calculating the exact value, make a reasoned estimate of whether at $2:00$ in the morning ($t = 2$) the temperature will be positive or below zero, comparing the weight of the constant term against the other terms.
2. b) Check your estimate by calculating the exact numerical value of $T(2)$ and assess whether the result is consistent with your prediction.
3. c) A report states that at $5:00$ in the morning ($t = 5$) the thermometer read exactly $0\text{ \textdegree C}$. Reason whether the value $t = 5$ is a root of the polynomial $T(t)$ and determine whether the report's statement is true or false.

Design the algebraic expression of a polynomial $P(x)$ that simultaneously satisfies these four conditions:

1. It has degree $3$ and its leading coefficient is $-2$.

2. It is an incomplete polynomial that lacks the degree $1$ term.

3. Its constant term is $8$.

4. The number $2$ is one of its roots.

1. a) Deduce step by step the value of each coefficient and write the polynomial ordered from highest to lowest degree.
2. b) Formally verify by calculating the numerical value that $x = 2$ is, indeed, a root of the polynomial you obtained.

Find the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 5x + 6$ is simultaneously divisible by $(x - 1)$ and $(x + 2)$.

In the design of a mechanical part, a right cylinder is inscribed inside a cone with base radius $R = 10$ cm and height $H = 15$ cm. If the radius of the base of the cylinder is $x$, express the volume of the cylinder as a polynomial in terms of $x$. What is the valid geometric domain for the variable $x$?

Perform the following operations with algebraic fractions and simplify the result fully: $\left( \frac{x + 2}{x - 2} - \frac{x - 2}{x + 2} \right) : \frac{4x}{x^2 - 4}$.

A third-degree polynomial $P(x)$ has roots $x = 1$, $x = -2$ and $x = 3$. Knowing that when dividing $P(x)$ by $x + 1$ the remainder is $24$, determine the expanded general expression of the polynomial $P(x)$.

Prove algebraically that the difference between the square of the sum of any two numbers $a$ and $b$ and the square of their difference is always equal to four times the product of those numbers.

Factorise the fourth-degree polynomial $P(x) = x^4 - 2x^3 - 7x^2 + 8x + 12$ using Ruffini's rule and solving quadratic equations. Clearly state all its roots.

The manufacturing cost of a fibre optic cable of length $x$ metres is given by a quadratic polynomial $C(x) = ax^2 + bx + c$. Knowing that manufacturing $10$ metres costs $150$ €, $20$ metres costs $380$ € and the fixed costs (manufacturing $0$ metres) are $40$ €, determine the expression for the polynomial $C(x)$ and calculate the exact cost of manufacturing $50$ metres.

In an architectural project, the shape of a parabolic arch is modelled using the polynomial $A(x) = -kx^2 + 6x$. Knowing that the arch has a maximum height at its vertex and that the polynomial is divisible by $x - 12$, find the value of $k$, factorise the polynomial and determine the total width of the arch at the base (the distance between the points where $A(x) = 0$).

The annual financial balance of an olive oil cooperative in Jaén (in thousands of euros) between the years 2018 and 2024 has been modelled using the polynomial $S(t) = t^3 - 8t^2 + 11t + 20$, where $t$ represents the number of years elapsed since 2018 (that is, $t = 0$ corresponds to 2018, $t = 1$ to 2019, etc.).

1. a) In the socioeconomic context of the cooperative, explain what practical meaning it has for a value of $t$ to be a root of the polynomial $S(t)$.
2. b) Management states in its report that in both 2022 ($t = 4$) and 2023 ($t = 5$) the cooperative reached the budgetary break-even point (balance exactly equal to zero). Algebraically check whether $t = 4$ and $t = 5$ are roots of $S(t)$ and determine whether the report's statement is strictly true in both cases.
3. c) If you calculate the numerical value for $t = -1$, you will get $S(-1) = 0$. Reason whether, from the perspective of the applied real-world model, it makes sense to interpret $t = -1$ as a valid root for managing the company.

The sustainability department of a town council analyses the monthly energy output of a community solar installation using the polynomial $R(m) = 2m^3 - 11m^2 + 17m - 6$, where $m \in \{1, 2, 3, \dots, 12\}$ represents the month of the year (January $= 1$, February $= 2$, etc.). The goal is to find out in which months the output drops to zero (that is, which values of $m$ are roots of $R(m)$).

Two technicians propose different strategies:

• Technique A (Lucía): Proposes systematically calculating the numerical value $R(m)$ by substituting all $12$ months of the year, one by one, until finding which ones yield $0$.

• Technique B (Mateo): Recalls that the integer roots of a polynomial with integer coefficients must be divisors of its constant term. Since the constant term is $-6$, he proposes checking only the values of $m$ that are positive divisors of $6$ contained within the months of the year.

1. a) Compare the mathematical validity and practical efficiency of both methods. Justify which of the two approaches is the most advisable to solve the problem without doing unnecessary calculations.
2. b) Apply the chosen method to check which of those candidate months are actually roots of $R(m)$ and interpret the result obtained, indicating the specific months in which the output drops to zero.