Exercises: Polynomials

Let $P(x) = 2x^3 - 5x^2 + 3x - 1$ and $Q(x) = -x^3 + 4x^2 - 2x + 5$ be polynomials. Calculate the polynomial resulting from the operation $2P(x) - Q(x)$ and express the result ordered in descending order of degree.

Perform the following polynomial division and clearly indicate the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$:

$(x^4 - 3x^3 + 2x^2 - x + 5) : (x^2 - x + 1)$

Use Ruffini's rule to divide the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by the binomial $x + 2$. State the quotient and the remainder.

Apply the Remainder Theorem to calculate the remainder of the division of the polynomial $P(x) = x^5 - 3x^4 + 2x^2 - 5x + 8$ by $x - 3$, without performing the full division.

Using the Factor Theorem, check whether the values $x = 2$ and $x = -1$ are roots of the polynomial $P(x) = x^3 - 2x^2 - x + 2$. Justify your answer by showing your calculations.

Factorise out the common factor and use algebraic identities to fully factorise the polynomial $P(x) = 3x^3 - 12x$.

Simplify the following algebraic fraction as far as possible, factorising the numerator and the denominator first:

$\frac{x^2 - 9}{x^2 + 6x + 9}$

Perform the following multiplication of algebraic fractions and simplify the result:

$\frac{2x}{x-1} \times \frac{x^2 - 1}{4x^2}$

Cian has to calculate the quotient and the remainder of the division of the polynomial $P(x) = 2x^3 - 5x + 3$ by $x + 2$. On his exam, he writes the following Ruffini table: he places the coefficients $2$, $-5$, and $3$ in the top row, and in the left-hand side box he places the number $2$. After carrying out the operations, he concludes that the remainder of the division is $9$ and that the quotient is $C(x) = 2x + 1$.

1. a) Identify and explain with reasoning the two conceptual errors that Cian made when setting up Ruffini's rule.
2. b) Correct the setup by carrying out the division properly and clearly write the quotient polynomial $C(x)$ and the remainder $R$.
3. c) Verify that the obtained remainder is correct by evaluating the polynomial using the remainder theorem.

To calculate the remainder of the division of the polynomial $P(x) = 3x^{50} - 4x^{25} + 2x - 1$ by the binomial $x - 1$, three students in the class propose different strategies:

• Aoife proposes using traditional Euclidean long division.

• Seán suggests using Ruffini's rule, filling in the intermediate-degree terms with zeros.

• Niamh claims that applying the remainder theorem is much more direct.

1. a) Compare the three methods with reasoning and explain why two of them are completely unfeasible or inefficient in this specific case.
2. b) Following the most appropriate method, calculate the remainder of the division.
3. c) From the value of the obtained remainder, deduce with justification whether the binomial $(x - 1)$ is a factor of the polynomial $P(x)$ or not.

The daily profits of a company, in thousands of euros, are modelled by the polynomial $B(x) = -x^3 + ax^2 + bx + 4$, where $x$ is the day of the week (from 1 to 7). It is known that on day 1 the profit is zero. Furthermore, when dividing $B(x)$ by $x-2$, the remainder is $2$. Calculate the parameters $a$ and $b$, and factorise the polynomial to determine on which other day of the week the profit is exactly zero again.

A packaging company designs open-topped rectangular boxes from a sheet of cardboard measuring $30 \text{ cm}$ by $20 \text{ cm}$. To do this, they cut squares of side $x$ from each corner and fold up the edges. Express the volume $V(x)$ of the box as a polynomial in standard expanded form. Then, calculate the exact volume if squares of $x = 4 \text{ cm}$ are cut out.

In an electrical circuit, the equivalent resistance $R_{eq}$ is given by the following algebraic fraction expression:

$R_{eq}(x) = \frac{2}{x^2-1} - \frac{1}{x-1} + \frac{x}{x+1}$

Perform the indicated operations, calculate the lowest common multiple of the denominators, and simplify the resulting fraction as far as possible. What is the value of the resistance if $x = 3 \ \Omega$?

Expand the power of the binomial $(2x - y)^4$ fully, using the Binomial Theorem or Pascal's triangle.

In a study of genetic combinations, a certain probability distribution is related to the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^6$. Find the exact numerical value of the constant term (the term independent of $x$) in this expansion.

Fully factorise the fourth-degree polynomial $P(x) = x^4 - 5x^2 + 4$. From its factorised form, state all its real roots.

In the division of the polynomial $P(x) = 2x^4 - x^3 + kx^2 - 3x + 5$ by $Q(x) = x^2 - x + 2$, find the value of the parameter $k$ so that the remainder of the division is a constant number (that is, a polynomial of degree 0). What is the exact value of this remainder?

The area of a specially designed window in a building is given by the expression $A(x) = \frac{x^3 - 8}{x^2 - 4}$. Simplify this algebraic fraction by factorising the numerator (difference of two cubes) and the denominator first. Then, evaluate the area when $x = 3$ metres.

Design a monic polynomial $P(x)$ of the lowest possible degree that simultaneously satisfies the following four conditions:

1. It admits $x = -2$ as a root of multiplicity 2 (double root).

2. Its constant term is $-12$.

3. When divided by $(x - 1)$, the remainder obtained is exactly $-18$.

4. It has no more real roots than those strictly necessary to satisfy the previous conditions.

1. a) Justify why the polynomial sought cannot be of degree 2 and deduce what its minimum degree must be.
2. b) Determine the algebraic expression of $P(x)$ both in its factored form and in its expanded form arranged from highest to lowest degree.
3. c) Explicitly verify that the polynomial found satisfies each and every one of the conditions stated.

Consider the polynomial $P(x) = x^4 - 3x^3 + 5x - 40$ and the divisor $d(x) = x - 4$.

1. a) Before calculating the exact result, make a reasoned estimate of the remainder's value by analyzing the order of magnitude of the two terms of highest degree when $x = 4$. Do you expect the remainder to be positive or negative? On the order of tens or hundreds?
2. b) Calculate the exact remainder of the division by applying the remainder theorem and critically evaluate whether your initial estimate was accurate.
3. c) Apply Ruffini's rule to obtain the quotient polynomial $C(x)$ and verify the fundamental division identity, $P(x) = (x - 4)\times C(x) + R$, evaluating both sides for the numerical value $x = 1$.

The thermal expansion curve of a new polymer is approximated by a cubic polynomial $E(t) = at^3 + bt^2 + ct + d$. In laboratory tests, it was found that at temperatures $t = 1 \ ^\circ\text{C}$, $t = 2 \ ^\circ\text{C}$, and $t = -1 \ ^\circ\text{C}$, the expansion is zero. However, at $t = 0 \ ^\circ\text{C}$, the expansion is $2 \text{ mm}$. Determine the value of the coefficients $a, b, c, d$ of the polynomial.

The deflection curve of a beam under load is modelled by the polynomial $y(x) = x^4 + ax^3 + bx^2 + cx + d$. It is known that the polynomial is exactly divisible by $(x-1)^2$. Furthermore, when divided by $x+1$ the remainder is $-8$, and when divided by $x-2$ the remainder is $4$. Set up the corresponding system of equations and find the values of $a, b, c$ and $d$.

In the design of a gear mechanism, the gear ratio is modelled by the following complex algebraic expression:

$T(x) = \frac{\frac{x^2 - 4}{x^3 + 8}}{\frac{x^2 - 2x}{x^2 - 2x + 4}}$

Simplify $T(x)$ as far as possible by factorising all numerators and denominators first. What mathematical (and therefore physical) problem occurs with the gear ratio if $x = 2$?

In the expansion of the binomial $(x + 2)^n$, it is known that the coefficient of the term containing $x^{n-2}$ is exactly $180$. Set up the corresponding equation using the Binomial Theorem, find the value of the natural number $n$, and calculate the constant term of this expansion.

A cylindrical container is inscribed inside a sphere of radius $R = 10 \text{ cm}$. The volume of the cylinder as a function of its semi-height $x$ is $V(x) = 2\pi x (100 - x^2)$. It is desired that the volume be exactly $768\pi \text{ cm}^3$. Formulate the polynomial equation $P(x) = 0$, find its integer root using Ruffini's rule, and factorise the resulting polynomial to find all possible semi-heights $x$ that satisfy the geometric condition.

Using the Factor Theorem, prove that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \ge 1$. Then, apply this property to find the obvious root and fully factorise the polynomial $P(x) = x^5 - 32$ over the set of real numbers.

Given the algebraic fraction that models the signal-to-noise ratio in a telecommunications network:

$S(x) = \frac{x^4 - 2x^3 - 7x^2 + 8x + 12}{x^3 - 5x^2 + 2x + 8}$

Fully factorise the numerator and the denominator using Ruffini's rule and by solving quadratic equations. Simplify the fraction as far as possible and determine, if it exists, for which real value of $x$ the simplified fraction is equal to $3$.

In a 3D design project, a solid cube with edge length $x+2 \text{ cm}$ is used as a starting point. Its interior is hollowed out by removing another concentric cube of edge length $x-1 \text{ cm}$. Express the volume of remaining material as a polynomial in standard form (expanding the cubed binomials and simplifying). Then, find the value of $x$ if it is known that the volume of remaining material is exactly $117 \text{ cm}^3$.

The volume of dammed water in a reservoir in the Shannon basin over a 12-month hydrological year ($t \in [0, 12]$, where $t = 0$ represents October 1) is modeled using the cubic polynomial $V(t) = -t^3 + b t^2 + c t + d$, expressed in cubic hectometers ($\text{hm}^3$). Official data from the ESB record the following:

• At the beginning of the hydrological year ($t = 0$), the reservoir contained $180\text{ hm}^3$.

• At the end of December ($t = 3$), the volume was once again exactly $180\text{ hm}^3$.

• At the end of March ($t = 6$), after the late-winter rains, the stored volume reached $288\text{ hm}^3$.

1. a) Translate the information into a system of equations and calculate the coefficients $b$, $c$, and $d$ with reasoning.
2. b) Using Ruffini's rule, decompose the polynomial $V(t) - 180$ into a product of three linear factors and determine in which other month of the hydrological year the reserves return exactly to the initial level of $180\text{ hm}^3$.
3. c) Interpret the sign of the factored polynomial $V(t) - 180$ on the interval $[0, 12]$ to indicate in which period of the year the reserves are above $180\text{ hm}^3$, and reason whether the model predicts a total emptying of the reservoir ($V(t) \le 0$) at any point.

A polynomial $P(x)$, whose degree is not specified a priori, produces a remainder equal to $-4$ when divided by $(x - 2)$, and a remainder equal to $8$ when divided by $(x + 1)$.

1. a) If $P(x)$ is divided by the quadratic polynomial $Q(x) = (x - 2)(x + 1) = x^2 - x - 2$, formally justify, using the fundamental Euclidean division identity $D(x) = d(x)\times C(x) + R(x)$, what the maximum degree that the remainder $R(x)$ can have is.
2. b) Write down the general algebraic expression of said remainder $R(x)$ and calculate its exact coefficients by applying the remainder theorem, proving why it is not necessary to know the global formula of $P(x)$ to determine it.
3. c) If the additional information is given that $P(x)$ is monic, of degree 3, and its constant term is $6$ (that is, the remainder when divided by $x$ is $6$), reconstruct the complete expression of $P(x)$ and factor it in $\mathbb{R}$, finding all its roots.