Exercises: Factorisation and Ruffini's rule

Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. Clearly state the quotient polynomial and the remainder of the division.

Without performing the division, use the remainder theorem to calculate the remainder obtained when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.

Use the factor theorem to check whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your calculations.

Factorise out the common factor in the following polynomial, simplifying the expression as much as possible:

$P(x) = 15x^5 - 10x^4 + 20x^3$

Factorise the following polynomials by recognising algebraic identities (square of a sum, square of a difference, or difference of two squares):

1. a) $A(x) = x^2 - 14x + 49$
2. b) $B(x) = 9x^2 - 16$

Given that $x = 1$ is one of the roots of the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$, use Ruffini's rule to factorise it and find the other two roots.

Given the already factorised polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly all of its roots and specify the multiplicity of each root (whether they are simple, double, etc.).

Construct a polynomial of degree 3 that has roots $x_1 = -2$, $x_2 = 0$ and $x_3 = 4$, and whose leading coefficient is $1$. Express the final result in expanded form (by multiplying out the factors).

Mateo has to determine whether the binomial $(x + 2)$ is a factor of the polynomial $P(x) = x^3 - 3x + 2$. To check this, he decides to apply Ruffini's rule and writes only the numbers $1$, $-3$, and $2$ in the top row of his table. In the left side cell, he places the value $-2$. After operating, he obtains a non-zero remainder and states: "Since the remainder is not zero, $(x + 2)$ does not divide the polynomial and, therefore, is not a factor."

1. a) Identify the two mistakes made by Mateo (one procedural when preparing the table and another in his conclusion).
2. b) Explain why it is essential to include a zero in place of missing terms when applying this method.
3. c) Perform the division correctly using Ruffini's rule and determine with justification whether $(x + 2)$ is actually a factor of $P(x)$, stating the quotient obtained.

To factor the polynomial $Q(x) = 3x^4 - 12x^2$, two classmates propose two different strategies:

• Alba's strategy: Find the integer divisors of the constant term and successively apply Ruffini's rule.

• Bruno's strategy: First factor out the greatest possible common factor and then apply special products.

1. a) Explain what insurmountable drawback Alba encounters if she tries to apply her strategy as she has formulated it.
2. b) Justify why Bruno's proposal is faster, safer, and more appropriate for this type of polynomial.
3. c) Apply Bruno's method to completely factor $Q(x)$, deducing all its real roots and the multiplicity of each.

A water tank is in the shape of a rectangular prism. Its volume as a function of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of the tank is exactly $x$, factorise the polynomial to find the algebraic expressions corresponding to the dimensions of its rectangular base.

Find the value of the parameter $k$ so that the polynomial $P(x) = x^3 + kx^2 - 3x + 8$ gives a remainder of $10$ when divided by $(x - 2)$.

The height of a roller coaster carriage, in metres, relative to time $t$ in seconds from the start of a drop, is modelled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factorise the polynomial to determine the exact times at which the carriage is at ground level ($H(t) = 0$).

Fully factorise the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$. To do this, first factorise out the common factor and then find the roots of the resulting polynomial.

Calculate the value of $m$ so that the polynomial $P(x) = x^3 - 2x^2 + mx + 6$ is exactly divisible by $(x - 3)$. Once the value of $m$ is found, fully factorise the polynomial $P(x)$.

The number of visits to a website (in thousands) during month $x$ since its launch is given by the expression $V(x) = x^3 - 14x^2 + 49x$. Factorise the expression to find in which month or months the number of visits was exactly zero.

Divide the polynomial $P(x) = 4x^3 - 8x^2 + 5x - 1$ by the binomial $(2x - 3)$. Hint: You can use Ruffini's rule by dividing by $(x - \frac{3}{2})$ and then adjusting the quotient obtained.

A packaging company designs a cardboard box whose volume is given by the polynomial $V(x) = x^3 - 12x^2 + 36x$. Factorise the polynomial fully to express the volume as the product of three linear dimensions (length, width and height).

We want to factor the polynomial $P(x) = x^3 + 4x^2 - 11x - 30$.

1. a) List all the candidates for integer roots from the divisors of the constant term.
2. b) Estimate before calculating: Observe the candidates with the largest absolute value ($\pm 15$ and $\pm 30$). Without applying Ruffini's rule, mentally compare the magnitude of $x^3$ against the sum of the remaining terms and explain why it is extremely unlikely that numbers so far from the origin are roots of this polynomial. In what interval would you expect to find the roots?
3. c) Use the remainder theorem evaluating only the candidates closest to zero ($\pm 1, \pm 2, \pm 3$) to find a first root without having to draw unnecessary Ruffini tables.
4. d) Complete the factorization of $P(x)$ using Ruffini's rule starting from the root found, state all real roots, and reflect on how the initial estimation helps save fruitless calculations.

A logistics company in Zaragoza operating at the PLAZA platform designs intermodal transport containers for rail. The payload volume of each container (in cubic metres, $\text{m}^3$) is given by the polynomial $V(x) = 2x^3 - 3x^2 - 11x + 6$, where the variable $x$ is a technical design dimension in metres, subject to the condition $x > 3$.

1. a) Apply Ruffini's rule to decompose $V(x)$ into the product of three first-degree binomials, which will represent the three spatial dimensions of the container (length, width, and height).
2. b) If the height of the container is represented by the expression $(2x - 1)\text{ m}$, what are the algebraic expressions of the other two dimensions that form the base?
3. c) If the engineers set the parameter to $x = 4\text{ m}$, calculate the actual metric dimensions of the container, its total volume in $\text{m}^3$ and in litres, and check that you obtain the same result using both the expanded and factored polynomial expressions.

A team of engineers analyses the stress distribution in a metal beam, which is modelled by the polynomial $P(x) = x^4 - x^3 - 7x^2 + x + 6$. Find all the roots of the polynomial to determine the exact points where the stress is zero.

Determine the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 4x + 4$ is simultaneously divisible by $(x - 1)$ and by $(x + 2)$.

A solar panel is rectangular in shape and its area is given by the polynomial $A(x) = x^4 - 81$. Fully factorise the polynomial into irreducible polynomials with real coefficients to find the possible algebraic expressions for its base and height.

Construct a polynomial $P(x)$ of degree 3 that has roots $x_1 = 1$, $x_2 = -2$ and $x_3 = 3$, knowing also that its value at $x = 0$ is $12$ (that is, $P(0) = 12$). Express the final polynomial in expanded form.

Fully factorise the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$. Note that it may contain multiple roots. Clearly state its roots and the multiplicity of each one.

Using polynomial factorisation and algebraic identities, prove algebraically that the difference between the squares of two consecutive odd numbers, $(2n+1)^2 - (2n-1)^2$, is always a multiple of $8$ for any integer $n$.

It is known that the polynomial $P(x) = 2x^3 + x^2 - 13x + 6$ has at least one integer root. Use the remainder theorem or Ruffini's rule to check that $x = 2$ is one of its roots. Next, fully factorise the polynomial to find its other two roots (one of which is a fraction).

In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Fully factorise the polynomial to find all the roots and their respective multiplicities, thereby deciphering the code.

Find the analytic expression of a degree 4 polynomial $P(x)$ that simultaneously satisfies the following four conditions:

1. Its leading coefficient is equal to $2$.

2. It has $x = -1$ as a root of multiplicity 2.

3. It is exactly divisible by the binomial $(x - 2)$.

4. When dividing $P(x)$ by $(x + 2)$, the remainder obtained is $-48$.

To solve it with justification:

1. a) Write the general factored form of $P(x)$ leaving an unknown fourth root $r$ indicated.
2. b) Apply the remainder theorem with condition 4 to set up a linear equation and solve for the exact value of the root $r$.
3. c) Write the complete, final factorization of $P(x)$ and then perform the necessary algebraic operations to express it in expanded form (in descending powers of $x$).

Consider the polynomial $P(x) = 2x^3 + kx^2 - 13x + 6$, where $k$ represents an unknown real parameter. It is known that $x = 2$ is one of its roots. To find the value of $k$ and complete the factorization, the mathematics department compares two didactic alternatives:

• Method 1 (Ruffini with parameter): Place the parameter $k$ directly in the Ruffini table dividing by $(x - 2)$, obtain the algebraic expression of the remainder, and set it equal to zero.

• Method 2 (Remainder theorem): Use the condition $P(2) = 0$ via direct numerical substitution to solve for $k$, and only perform Ruffini's rule once all coefficients are known.

1. a) Compare both alternatives: explain what algebraic difficulties Method 1 presents and why Method 2 is more efficient and carries less risk of error in 4th year of ESO.
2. b) Apply Method 2 to calculate the exact value of $k$.
3. c) Once $k$ is determined, factor the polynomial $P(x)$ completely. Justify obtaining the fractional root using the rational root theorem ($p/q$) and write the final factorization, making sure not to omit any coefficient.

Unit Challenge. Imagine you are planning a mission with an environmental drone to measure urban emissions using real meteorological data.

1. Data and vector path: Check AEMET for today's average wind speed ($v$) and temperature in your municipality. Define the take-off vector $\vec{u}=(v, 2)$ and the wind drift vector $\vec{w}=(-1, 3)$. Perform the combined operation $2\vec{u} + \vec{w}$ to calculate the drone's final displacement vector.

2. Battery life: Simulate battery consumption in GeoGebra or a spreadsheet using the exponential function $B(t) = 100 \cdot 0,85^t$ (with $t$ in flight hours). Estimate after how many hours the charge drops below $15\,\%$.

3. Performance: Aerodynamic drag depends on altitude $x$ according to the algebraic fraction $F(x) = \frac{x^3 - 3x^2 - 4x + 12}{x^2 - 5x + 6}$. Apply Ruffini's rule to factor the numerator and denominator, simplify the expression, and identify which critical altitudes make the denominator zero.

4. Final report: Write a one-page report containing the vector trajectory, the battery graph, and the altitudes to be avoided based on the simplified fraction.

Evaluation criteria:

• Rigor in factoring via Ruffini's rule and simplifying the algebraic fraction.

• Accuracy in vector operations and use of real AEMET data.

• Correct modelling and interpretation of the exponential function with digital tools.