Exercises: Operations with polynomials

Given the polynomials $P(x) = 3x^3 - 2x^2 + 5x - 1$ and $Q(x) = -x^3 + 4x^2 - 5x + 3$, calculate the expressions resulting from the operations $P(x) + Q(x)$ and $P(x) - Q(x)$.

Perform the multiplication of the monomial $-4x^2$ by the polynomial $2x^3 - \frac{1}{2}x^2 + 3x - 5$ and write the result in descending order of degree.

Calculate the product of the following polynomials and collect like terms: $(2x - 3) \cdot (x^2 + 4x - 1)$.

Expand the following expression using the standard identity formula for the square of a sum: $(3x + 5y)^2$.

Apply the corresponding algebraic identity to expand the square of this difference: $(\frac{1}{2}x^2 - 4)^2$.

Calculate the result of the following operation using the product of a sum and a difference identity: $(7a + 2b) \cdot (7a - 2b)$.

Add the polynomials $A(x) = x^2 - 3x + 2$ and $B(x) = 2x^2 + x - 5$. Once you have obtained the resulting polynomial, calculate its numerical value for $x = -2$.

When multiplying the polynomial $A(x) = 5x^4 - 3x + 2$ by the polynomial $B(x) = -2x^3 + x^2$, state the degree of the resulting polynomial and the value of its leading coefficient, without performing the full multiplication.

Elena needs to calculate the leftover material in a cutting process, and to do so she must simplify the algebraic expression $(3x - 2)^2 - (4x^2 - 6x + 7)$. On her exam, she writes the following working:

• Step 1: $(3x - 2)^2 = 9x^2 - 4$

• Step 2: $9x^2 - 4 - 4x^2 - 6x + 7$

• Step 3: Final result $= 5x^2 - 6x + 3$

1. a) Identify and explain, with reasoning, the two conceptual errors that Elena made (one in expanding the special product and another when removing the parentheses preceded by the minus sign).
2. b) Carry out the correct step-by-step working until you obtain the simplified polynomial.
3. c) Check that your solution is consistent by substituting $x = 0$ both into the initial expression and into your final polynomial.

The municipal electric bicycle rental service of a Spanish city bills each user monthly based on the additional minutes consumed, represented by the variable $x$.

• The basic subscription fee follows the polynomial $C(x) = 12 + 0,10x$ euros.

• The insurance and maintenance surcharge is $S(x) = 5 + 0,05x$ euros.

• As a sustainable mobility incentive, the city council applies a direct discount expressed by $D(x) = 3 + 0,02x$ euros.

1. a) State the polynomial operation used to calculate the final monthly cost $G(x) = C(x) + S(x) - D(x)$.
2. b) Remove the parentheses, change the corresponding signs, and combine like terms to obtain the simplified expression for $G(x)$.
3. c) If a person has recorded $x = 100$ additional minutes in the month, calculate how much they will pay in total and interpret what real-life meaning the constant term of the polynomial you obtained has.

A rectangular garden has dimensions that depend on a value $x$. Its length is $3x + 2$ metres and its width is $x + 5$ metres. Express the total area of the garden as a polynomial in $x$ and calculate how many square metres it will have if $x = 4$.

An open-topped box is made from a rectangular sheet of cardboard measuring $30 \text{ cm}$ long by $20 \text{ cm}$ wide. To do this, four identical squares of side $x$ are cut from the corners and the flaps are folded up. Express the volume of the resulting box as a polynomial in the variable $x$.

An electronics factory incurs fixed monthly costs of $500$ euros. In addition, the production cost per batch of parts is $40 - 2x$ euros, where $x$ represents the number of batches. If exactly $x^2 + 5x$ batches are produced in a month, express the factory's total monthly cost as a polynomial in $x$.

Expand and simplify fully the following algebraic expression by applying the corresponding standard identities:

$(2x + 3)^2 - (2x - 3)^2 + (x - 1) \cdot (x + 1)$

Given the polynomials $A(x) = x^3 - 4x^2 + x - 1$ and $B(x) = 2x^3 + x^2 - 5$, determine the expression for an unknown polynomial $P(x)$ that satisfies the following equality:

$2 \cdot A(x) - P(x) = B(x)$

When carrying out the multiplication of the polynomials $P(x) = x^2 + kx - 3$ and $Q(x) = 2x^2 - 5x + 1$, it is observed that the coefficient of the term of degree $2$ in the resulting polynomial is $-13$. Set up the corresponding equation and determine the value of the parameter $k$.

In a physics experiment, the height in metres of a projectile $A$ is given by the polynomial $H(t) = -5t^2 + 20t + 2$, and that of a projectile $B$ by $G(t) = -5t^2 + 15t + 10$, where $t$ is the elapsed time in seconds. Find the polynomial expressing the difference in height between projectile $A$ and projectile $B$, and determine the exact moment at which both projectiles are at the same height.

Observe the following figure representing a square of side $a+b$ subdivided into four rectangular regions. Write down the sum of the areas of the four regions and geometrically prove the expansion of the standard identity $(a+b)^2$.

To simplify the expression $E(x) = (3x + 4)^2 - (3x - 4)^2$, two classmates propose different strategies:

• Lucía proposes expanding each term separately using the special products for the square of a sum and the square of a difference, and then subtracting the resulting polynomials.

• Mateo proposes interpreting the entire expression as a difference of squares of the form $A^2 - B^2 = (A + B) \cdot (A - B)$, taking $A = 3x + 4$ and $B = 3x - 4$.

1. a) Apply Lucía's method, clearly writing the intermediate expansions and the final subtraction of the terms.
2. b) Apply Mateo's method, substituting $A$ and $B$ into the sum-and-difference formula and simplifying the factors before multiplying.
3. c) Compare both approaches: do you get the same result? Explain with reasoning which of the two methods seems faster to you and in which one you think there is a lower probability of making mistakes with signs.

On an agricultural farm in Castilla-La Mancha, they are going to install solar panels on the roof of an industrial building for self-consumption. The rectangular area intended for the panels has a width of $(x + 3)$ metres and a length of $(3x + 5)$ metres, where $x$ is a measurement in metres associated with the required power.

1. a) Obtain the expanded polynomial $S(x)$ that represents the total area in square metres by multiplying both dimensions.
2. b) Estimate mentally: if setting $x = 10\text{ m}$ is required to cover the building's consumption, round each dimension mentally to the nearest ten and predict whether the area will be closer to $300\text{ m}^2$, $450\text{ m}^2$, or $600\text{ m}^2$. Explain how you made your rounding.
3. c) Calculate the exact numerical value of $S(10)$ by substituting $x = 10$ into the expanded polynomial from section a), and assess whether your previous estimate was accurate by calculating the difference between the exact value and the estimated one.

Find all possible values for the parameters $a$, $b$ and $c$ such that the following polynomial equality holds for any real number $x$:

$(ax + b)^2 + cx = 9x^2 - 24x + 16$

In a rectangular metal plate whose dimensions are $4x + 6$ and $2x + 1$ millimetres, three identical circular holes of radius $r = x$ are drilled. Express the area of the resulting metal plate, once the holes have been made, as a polynomial in $x$ (leave the number $\pi$ indicated in the coefficients that require it).

Prove algebraically that the product of the sum of two variables and their difference, multiplied by the sum of the squares of those same variables, is equal to the difference of their fourth powers. To do so, expand and simplify step by step the expression: $(x - y) \cdot (x + y) \cdot (x^2 + y^2)$.

A software company estimates that if it sells its new application at a price of $p(x) = 120 - x$ euros, it will manage to sell $q(x) = 500 + 10x$ licenses. Knowing that total revenue is calculated by multiplying the price by the quantity sold, and that the company's fixed and variable costs amount to $C(x) = 2000 + 50x$ euros, find the profit polynomial $B(x)$ in its fully simplified form.

Consider the polynomial $P(x) = (2x^2 - 3x + 2)^3 \cdot (x^2 - x + 1)^2$. Knowing that the sum of all the coefficients of a polynomial (including the constant term) can be obtained by evaluating said polynomial at $x = 1$, calculate the exact sum of the coefficients of the resulting polynomial that would be obtained by fully expanding the expression for $P(x)$.

The polynomial $M(x) = x^4 - 2x^3 + ax^2 + bx + c$ is the exact result of squaring the polynomial $N(x) = x^2 - x - 3$. Perform the operation $N(x)^2$, collect like terms, and determine the corresponding numerical values for the parameters $a$, $b$ and $c$.

A solid cylinder of radius $r = x+1$ and height $h = 3x$ is inserted vertically inside a hollow square-based prism, whose base side measures $L = 2x+2$ and has the same height $3x$. Express as a polynomial in $x$ the volume of the space left free between the cylinder and the walls of the prism.

A magician asks a spectator to think of any number $x$. Next, he gives the following instructions: "Add 3 to your number, square the result, subtract the square of the original number, subtract 9, and finally divide everything you have left by 6." Prove step by step, using operations with polynomials, that the final result of this trick is always exactly the original number $x$ that the spectator thought of.

Reconstruct two first-degree polynomials with integer coefficients, $P(x)$ and $Q(x)$, from the following clues about their operations:

1. Their product is the second-degree polynomial: $P(x) \cdot Q(x) = 2x^2 + 5x - 3$.

2. Their sum is the polynomial: $P(x) + Q(x) = 3x + 2$.

3. The leading coefficient (the $x$ term) of $P(x)$ is strictly greater than the leading coefficient of $Q(x)$.

1. a) Deduce what the degree 1 coefficients of both polynomials must necessarily be by analysing how the term $2x^2$ is generated in the multiplication and the term $3x$ in the sum.
2. b) Analise the possible pairs of integers whose product is $-3$ and check which of them simultaneously satisfies that their sum is $+2$ and that the cross product generates the middle term $+5x$.
3. c) Write the final expressions for $P(x)$ and $Q(x)$, and calculate the subtraction $P(x) - Q(x)$, expressing it in its simplified form.

An artisanal bakery SME in Toledo is studying the economic feasibility of its bakehouse. The monthly costs and revenues depend on the number of weekly batches produced, represented by $x$ (with $x \ge 10$):

• Raw material costs (organic flours, seeds, yeasts): $M(x) = 2x^2 + 30x + 180$ euros.

• Energy cost and maintenance of wood-fired and electric ovens: $E(x) = x^2 + 20x + 120$ euros.

• Gross revenue from in-store sales and deliveries: $I(x) = 4x^2 + 10x - 200$ euros.

1. a) Obtain the simplified polynomial representing the total monthly costs, $C(x) = M(x) + E(x)$.
2. b) Model the monthly net profit using the subtraction $B(x) = I(x) - C(x)$, combining all like terms.
3. c) An advisor says: «Since the constant term of the profit is $-500$, the more the bakery produces, the greater the losses will be because that negative term always subtracts». Refute the advisor's statement algebraically: explain which term dominates when production $x$ is large, calculate for which exact number of weekly batches the net profit is zero ($B(x) = 0$), and explain from what production volume the business becomes profitable.