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:

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

The daily profits of a company, in thousands of pesos, 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 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 factoring the numerator (difference of two cubes) and the denominator first. Then, evaluate the area when $x = 3$ meters.

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 $117 \text{ cm}^3$.