Exercises: Operations with polynomials

Calculate the numerical value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 4$ for $x = -2$ and for $x = 1$.

Given the polynomials $A(x) = 3x^2 - 5x + 1$ and $B(x) = -x^2 + 4x - 6$, calculate the expression resulting from carrying out the operation $2 \cdot A(x) - B(x)$.

Multiply the polynomials $P(x) = x^2 - 3x + 2$ and $Q(x) = 2x - 5$. Express the result ordered from highest to lowest degree.

Divide the polynomial $D(x) = 4x^4 - 2x^3 + 6x^2 - x + 5$ by the binomial $d(x) = x^2 + 2$. Clearly state the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$.

Use Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by $(x - 2)$. Write down the quotient and remainder obtained.

Expand the following expressions using the appropriate algebraic identities:

1. a) $(3x + 4)^2$
2. b) $(2x - 5)^2$

Express the following binomials as a product of two factors using the difference of two squares identity:

1. a) $16x^2 - 25$
2. b) $9x^4 - 49$

Expand and fully simplify the following algebraic expression: $(x + 3)(x - 3) - (x - 2)^2$.

In a 4th year ESO exam, Marcos has to simplify the algebraic expression $A(x) = (3x - 2)^2 - (3x + 2)(3x - 2)$. On his answer sheet, he writes the following:

$A(x) = 9x^2 - 4 - (9x^2 - 4) = 9x^2 - 4 - 9x^2 - 4 = -8$.

1. a) Identify and explain in detail the two conceptual errors Marcos made (one in expanding the square of a difference and the other in the signs when removing the parentheses).
2. b) Carry out the correct expansion step by step using notable identities and calculate the numerical value of the resulting polynomial for $x = -1$.

You are going to calculate the product of the polynomials $P(x) = 2x^2 - 5$ and $Q(x) = 3x^3 + 4x - 1$.

1. a) Before carrying out the term-by-term multiplications, estimate with reasoning:

• What should be the degree of the resulting polynomial $P(x)\cdot Q(x)$?

• What should be its leading coefficient and its constant term? Justify how you can deduce these two values in a few seconds without expanding the entire operation.

1. b) Carry out the complete product, combine like terms, and check whether your initial estimation was accurate.
2. c) Assess the reasonableness of your final result by numerically calculating $P(1)$, $Q(1)$, and the value for $x = 1$ of the product polynomial obtained, verifying that the equality of the results holds.

A rectangular garden has dimensions of $(2x + 5)$ metres long and $(2x - 5)$ metres wide. Inside it, a square fountain with side length $(x - 1)$ metres will be built, and the rest will be covered with lawn. Express the area of the lawn area as a simplified polynomial in $x$.

An open-topped cardboard box is to be made from a rectangular sheet measuring $30 \text{ cm}$ long by $20 \text{ cm}$ wide. To do this, four identical squares with side length $x \text{ cm}$ are cut out from each corner and the flaps are folded upwards. Find the polynomial $V(x)$ that expresses the volume of the resulting box in terms of $x$.

Calculate the value of the parameter $k$ such that the division of the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$ by the binomial $(x + 2)$ has a remainder equal to $10$. Use the remainder theorem to justify your answer.

Expand and fully simplify the following algebraic expression, paying special attention to signs and algebraic identities:

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

The production cost of $x$ units of an electronic device is given by the polynomial $C(x) = 50x + 200$. The retail price of each unit depends on the quantity manufactured and follows the expression $p(x) = 300 - 2x$.

Find the profit polynomial $B(x)$, knowing that profit is equal to total revenue (quantity times price) minus total costs. What will the profit be if $40$ units are manufactured and sold?

Knowing that dividing a polynomial $P(x)$ by $(x - 3)$ gives a quotient of $C(x) = 4x^2 - 2x + 1$ and a remainder of $R = -7$, reconstruct the polynomial $P(x)$ and express it in expanded form.

The height in metres of a ball thrown vertically as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second ball, thrown from a different point, follows a trajectory described by $G(t) = -5t^2 + 15t + 10$.

Find the polynomial $D(t)$ representing the difference in height between the first and second balls ($D(t) = H(t) - G(t)$). At what instant $t$ are both balls at the same height?

Check whether the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by carrying out the corresponding division. If so, write the factorisation of $P(x)$ as the product of two polynomials.

We want to find the remainder of dividing the polynomial $P(x) = 4x^{20} - 3x^7 + 5x - 2$ by $(x + 1)$ and determine whether $(x + 1)$ is a factor of $P(x)$. Three members of a working team propose different approaches:

• Andrés says: "Let's do traditional long Euclidean division in a box."

• Bea says: "Let's apply Ruffini's rule by drawing the table with coefficients."

• Carlos says: "Let's use the remainder theorem directly by substituting $x = -1$."

1. a) Compare the three methods, explaining why the proposals of Andrés and Bea are extraordinarily inefficient and prone to counting errors in this specific exercise.
2. b) Justify which method is ideal and apply it to find the remainder of the division.
3. c) In light of the value obtained, conclude with reasoning whether the polynomial $P(x)$ is divisible by $(x + 1)$ and what geometric or algebraic meaning this fact has regarding the root $x = -1$.

An express courier company located in a logistics center in Zaragoza designs reinforced cardboard boxes for eco-friendly shipments. The interior dimensions of each box (in decimetres) are: length $(2x + 1)$, width $(x + 3)$, and height $(x - 1)$, with $x > 1$.

1. a) Model the total interior volume of the box in $\text{dm}^3$ using an expanded and ordered polynomial $V(x)$.
2. b) To seal the package, continuous tape is placed around the horizontal contour (perimeter of the base, formed by the length and width) and another around the vertical crosswise contour (perimeter formed by the width and height). Express the total tape length required (without overlaps) using a simplified polynomial $L(x)$.
3. c) If the design specifications set $x = 5\text{ dm}$, calculate the volume in litres ($1\text{ dm}^3 = 1\text{ litre}$) and the tape length in centimetres. Check whether the package complies with standard courier regulations in Spain, which require that the sum of its three dimensions does not exceed $25\text{ dm}$.

Find the values of the parameters $a$ and $b$ such that the polynomial $P(x) = x^3 + ax^2 + bx - 6$ is exactly divisible by $(x - 2)$ and, at the same time, leaves a remainder of $-12$ when divided by $(x + 1)$.

A running track consists of a central rectangular region of length $3x$ and width $2x$, capped at its ends by two semicircles of diameter $2x$. Express the total area and outer perimeter of the track as polynomials in terms of $x$ (leave the number $\pi$ indicated in the coefficients).

Prove algebraically that, for any real number $x > 1$, the expressions $a = x^2 - 1$, $b = 2x$ and $c = x^2 + 1$ always form the sides of a right-angled triangle (that is, they satisfy Pythagoras' theorem). Check your result by evaluating the polynomials for $x = 4$.

A startup estimates that its monthly revenue follows the polynomial $I(x) = -2x^3 + 75x^2$ and its operating costs are $C(x) = 15x^2 + 400x$, where $x$ is the number of months elapsed since its founding.

Find the profit polynomial $B(x)$. Knowing that in month $x = 10$ the profit is exactly zero (break-even point), factor the polynomial $B(x)$ to determine in which other month the profit becomes zero again.

Simplify the following algebraic expression, factorising the final result if possible:

$\frac{(2x - y)^3 - (2x + y)^3}{-2y}$

Next, evaluate the resulting polynomial for $x = 2$ and $y = -1$.

Determine the values of the coefficients $m$ and $n$ such that the division of the polynomial $D(x) = 4x^4 - 2x^3 + mx^2 + nx - 5$ by the divisor $d(x) = 2x^2 - x + 3$ is exact.

A water tank is cylindrical with a base radius of $(x + 2)$ metres and a height of $(2x - 1)$ metres. A solid cubic block with edge length $x$ metres is submerged inside it. Express, using an ordered polynomial, the volume of water needed to fill the remaining space in the tank. Leave the constant $\pi$ indicated in any coefficients that require it.

A polynomial $P(x)$ of degree 3 has roots $x = 1$, $x = -2$ and $x = 4$. Given also that the value of the polynomial for $x = 0$ is equal to $16$, find the expanded expression of $P(x)$ (that is, in the form $ax^3 + bx^2 + cx + d$).

A third-degree polynomial $P(x)$ has a leading coefficient of $3$. The following data about its divisions are known:

• It is exactly divisible by $(x - 2)$ and by $(x + 1)$.

• When dividing $P(x)$ by $(x - 1)$, the remainder obtained is $12$.

1. a) Using the remainder theorem and the fundamental relationship between divisors, factors, and roots, state the factored structure of $P(x)$ in terms of a single unknown root $c$, justifying why it is not necessary to set up a system of four linear equations with four unknowns.
2. b) Determine the exact value of the root $c$ from the remainder of the division by $(x - 1)$.
3. c) Write the expanded expression of $P(x)$ in the form $ax^3 + bx^2 + cx + d$ and, applying Ruffini's rule, calculate the remainder of dividing this polynomial by $(x + 2)$, verifying that it matches the numerical value $P(-2)$.

In an irrigation community in Castilla-La Mancha, an automated solar pumping system is installed. The annual maintenance cost (in euros) is modeled by the polynomial $C(x) = 2x^4 - 3x^3 + 5x^2 - 8$, where $x$ represents the number of active pumping sectors ($x \ge 2$). This cost is divided equally among the irrigation sectors, whose number is given by $S(x) = x - 2$.

1. a) Use Ruffini's rule to calculate the polynomial $Q(x)$ representing the annual cost per sector and the residual cost $R$ that remains unallocated in the community reserve fund. Remember to include all degrees in the table and write the relation $C(x) = S(x)\cdot Q(x) + R$.
2. b) A farmer claims that if the network is expanded to more sectors (increasing the value of $x$), the amount of the reserve fund $R$ will increase because the installation is larger. Refute his claim with reasoning using the formal definition of the division algorithm and the remainder theorem, explaining what degree $R(x)$ must necessarily have.
3. c) The community receives a subsidy from the Irrigation Modernization Plan that reduces the total cost according to the expression $A(x) = x^4 - (x^2 - 1)^2$. Simplify the subsidy polynomial $A(x)$ algebraically using notable identities and deduce the new undistributed residual remainder $R_{\text{nuevo}}$ after applying the aid, justifying the calculation using the remainder theorem without drawing the entire Ruffini table again.