Exercises: Polynomials

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

Perform the following polynomial division and clearly indicate which is 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 perform the division of the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by the binomial $x + 2$. Indicate 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 needing to perform the full division.

Using the Factor Theorem, check if 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 the calculations.

Factor out the common factor and use algebraic identities to completely factorize the polynomial $P(x) = 3x^3 - 12x$.

Simplify the following algebraic fraction as much as possible, previously factorizing the numerator and the denominator:

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

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

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

The daily profits of a company, in thousands of dollars, are modeled 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 factorize the polynomial to determine on which other day of the week the profit is exactly zero again.

A packaging company designs rectangular boxes without a lid from a cardboard sheet of $30 \text{ cm}$ by $20 \text{ cm}$. To do this, they cut out squares of side $x$ at each corner and fold the edges upward. Express the volume $V(x)$ of the box as a polynomial in standard form (expanded). 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 expression of algebraic fractions:

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

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

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

In a study of genetic combinations, the distribution of a certain probability 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 that does not depend on $x$) in this expansion.

Completely factorize the fourth-degree polynomial $P(x) = x^4 - 5x^2 + 4$. From its factorized form, indicate 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 that 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 previously factorizing the numerator (difference of two cubes) and the denominator. Then, evaluate the area when $x = 3$ feet.

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 has been verified 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 bending curve of a beam subjected to a load is modeled 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 dividing it by $x+1$ the remainder is $-8$, and when dividing it 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 transmission ratio is modeled 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 much as possible by previously factorizing all numerators and denominators. What mathematical (and therefore physical) problem occurs with the transmission 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 said 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 factorize the resulting polynomial to find all possible semi-heights $x$ that satisfy the geometric condition.

Prove, using the Factor Theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \ge 1$. Next, apply this property to extract the obvious root and completely factorize the polynomial $P(x) = x^5 - 32$ in the set of real numbers.

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

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

Completely factorize the numerator and the denominator using Ruffini's rule and solving quadratic equations. Simplify the fraction as much as possible and determine, if it exists, for what real value of $x$ the simplified fraction is equal to $3$.

In a 3D design project, we start with a solid cube of edge $x+2 \text{ cm}$. Its interior is hollowed out by extracting another concentric cube of edge $x-1 \text{ cm}$. Express the volume of the 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 the remaining material is exactly $117 \text{ cm}^3$.