Exercises: Ruffini's rule

Divide the polynomial $P(x) = 3x^3 - 2x^2 + 5x - 4$ by the binomial $x - 2$ using Ruffini's rule, and state the resulting quotient and remainder.

Apply Ruffini's rule to divide $P(x) = x^4 - 16$ by $x + 2$. Write down clearly the quotient polynomial and the value of the remainder.

Check, using Ruffini's rule, whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$.

Calculate the numerical value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 1$ for $x = -1$ by applying the Remainder Theorem using Ruffini's rule.

Determine the value of the constant $k$ so that the polynomial $P(x) = x^3 - kx^2 + 3x - 2$ is exactly divisible by $x - 1$.

Factorise the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$ knowing in advance that $x = 1$ is one of its roots.

Find the remainder of the division of the polynomial $P(x) = x^5 + 3x^4 - 2x^2 + 7$ by $x + 3$ using Ruffini's rule.

Determine, without performing the full division and using the Remainder Theorem or Ruffini's rule, whether the binomial $x + 1$ is a factor of the polynomial $P(x) = x^4 + x^3 - x^2 + x + 2$.

Mateo has to divide the polynomial $P(x) = 4x^3 - 6x^2 + 8x - 5$ by the binomial $2x - 1$. To do this, he applies Ruffini's rule using the value $a = 1/2$. In the last row of his table, he successively obtains the coefficients $4$, $-4$, $6$ and, as the last number, $-2$. From this, Mateo concludes in his exam: "The quotient of the division is $C(x) = 4x^2 - 4x + 6$ and the remainder is $R = -2$". Identify and explain with reasoning the conceptual error made by Mateo, prove using the division algorithm check that his quotient cannot be correct, and write the complete and corrected solution (actual quotient and remainder).

To calculate the numerical value of the polynomial $P(x) = 2x^4 - 5x^3 + 3x - 7$ at $x = 3$, two different approaches can be considered: substituting $x = 3$ directly and calculating each power, or applying Ruffini's rule by dividing by $x - 3$ to find the remainder.

1. a) Compare both methods by analyzing the type and number of operations that each one requires (high powers versus successive additions and multiplications) and justify which one is faster and less error-prone when a calculator is not available.
2. b) Apply Ruffini's rule (paying attention to missing terms) to obtain the remainder, deduce the value of $P(3)$, and state with reasoning whether $x - 3$ is a factor of $P(x)$.

The volume of a box in the shape of a right rectangular prism is given by the polynomial $V(x) = x^3 + 5x^2 + 2x - 8$. If the height of the box is known to be $x - 1$, determine the polynomial representing the base area $A(x)$.

The profits of a company, in thousands of euros, are modelled during its first years of operation by the polynomial $B(x) = -x^3 + 10x^2 - 31x + 30$, where $x$ represents time in years. Find the exact years in which the company's profit was zero (break-even points).

Divide the polynomial $P(x) = 2x^3 - 5x^2 + 3x - 1$ by the binomial $x - \frac{1}{2}$ using Ruffini's rule. Express the final result by stating the quotient and remainder.

Determine the values of the parameters $m$ and $n$ so that the polynomial $P(x) = x^4 - 3x^3 + mx + n$ is exactly divisible by $x - 2$ and, at the same time, gives a remainder of $4$ when divided by $x + 1$.

The position of a particle in straight-line motion is given by the equation $s(t) = t^3 - 7t^2 + 14t - 8$, where $t$ is the time in seconds. Determine the instants in time at which the particle passes through the origin of coordinates ($s(t) = 0$).

Calculate the numerical value of the polynomial $P(x) = x^4 - 2,5x^3 + 1,5x^2 - 3x + 2$ for $x = 2$ using exclusively the setup of Ruffini's rule.

The area of a rectangular solar panel is described by the polynomial $A(x) = 2x^3 + 7x^2 + 4x + 3$. If the width of the panel is known to be $x + 3$, find the polynomial representing its length.

Fully factorise the polynomial $P(x) = 2x^3 + 3x^2 - 8x + 3$ by finding all its roots.

In an efficiency study on a freight rail transport line between Madrid and Sevilla, the accumulated auxiliary fuel consumption (in liters) of a locomotive fits the polynomial $C(x) = 2x^3 - 8x^2 + kx - 10$, where $x$ is the elapsed time in hours ($x \in [3, 5]$). The engineers need to divide this total consumption by a linear load factor represented by $D(x) = 2x - 6$ to obtain the consumption per unit of load. It is known empirically that after $3$ hours of travel, the residual consumption not absorbed by the linear model (that is, the remainder of the division) is exactly $8$ liters.

1. a) Apply Ruffini's rule adapted for divisors of the form $cx - a$ to set up an equation and determine the value of the parameter $k$.
2. b) With the obtained value of $k$, find the polynomial that represents the actual quotient of the division.
3. c) Interpret whether at $3$ hours the load factor divides the accumulated consumption exactly and what practical meaning it has that the remainder is not zero.

Given the polynomial $P(x) = 2x^4 - 31x^2 - 15x + 8$ and the divisor $x - 4$:

1. a) Without carrying out Ruffini's table yet, make a prior estimate of the sign and order of magnitude of the remainder $P(4)$: mentally compare the predominant value of the leading term ($2 \cdot 4^4$) against the quadratic term ($-31 \cdot 4^2$). From this estimate, do you consider it plausible that $x - 4$ is an exact factor, or do you expect a remainder far from zero?
2. b) Apply Ruffini's rule completely to obtain the exact quotient and remainder, justifying the presence of zeros in the table if necessary.
3. c) Contrast the remainder obtained with your initial estimate and check the validity of the result by verifying the fundamental identity $P(x) = (x - 4) \cdot C(x) + R$ for the simple particular cases $x = 0$ and $x = 1$.

Find the values of $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 12x + 4$ is divisible by $(x - 2)^2$.

Perform the division of the polynomial $P(x) = 4x^3 - 8x^2 + 7x - 3$ by the binomial $2x - 3$ using Ruffini's rule. Note that you must adapt the divisor to apply the algorithm and subsequently adjust the quotient and remainder obtained.

The volume of a spherical cap is given by the formula $V = \frac{\pi h^2}{3}(3R - h)$, where $R$ is the radius of the sphere and $h$ is the height of the cap. If the radius of a sphere is $R = 5\text{ cm}$ and a cap with a volume of $V = \frac{52\pi}{3}\text{ cm}^3$ is required, set up the corresponding equation and use Ruffini's rule to find the height $h$.

Fully factorise the fifth-degree polynomial $P(x) = x^5 - 5x^4 + 7x^3 - 2x^2 + 4x - 8$ into irreducible polynomials over the real numbers.

Apply Ruffini's rule to divide the polynomial $P(x) = x^5 - 32$ by $x - 2$. From the pattern observed in the coefficients of the quotient, deduce and write down the general expression for the quotient polynomial when dividing $x^n - a^n$ by $x - a$.

The temperature of a chemical reaction as a function of time $t$ (in minutes) is modelled by the polynomial $T(t) = -t^4 + 8t^3 - 17t^2 + 10t$. Determine the times at which the temperature of the reaction is exactly $0^\circ\text{C}$.

Find the expression of a cubic polynomial $P(x)$ whose leading coefficient is $1$, knowing that it equals zero for $x = 1$ and $x = 2$, and that when divided by $x - 3$ it gives a remainder of $10$.

Solve the polynomial inequality $x^4 - 5x^3 + 5x^2 + 5x - 6 \ge 0$. To do this, factorise the polynomial first using Ruffini's rule and then analyse the sign in the corresponding intervals of the real line.

On the blackboard of a Bachillerato classroom, the solution to a problem in which two successive divisions were applied to a third-degree polynomial $P(x) = ax^3 + bx^2 + cx + d$ has been partially erased. Only the following legible notes remain:

1. The leading coefficient is $a = 2$.

2. When dividing $P(x)$ by $x + 1$ using Ruffini's rule, a remainder $R_1 = -12$ and a quotient polynomial $Q(x)$ were obtained.

3. When dividing said quotient $Q(x)$ by the binomial $2x - 1$ using the adaptation of Ruffini's rule, the division was exact (remainder $R_2 = 0$) and the final quotient was $x + 4$.

1. a) Reconstruct in reverse the algebraic expression of the intermediate quotient $Q(x)$, detailing how the divisor $2x - 1$ is involved.
2. b) Reconstruct the complete algebraic expression of the original polynomial $P(x)$.
3. c) Without calculating $P(-1)$ or $P(1/2)$ by evaluating the obtained formula, justify using the definition of division and the remainder theorem whether $x = -1$ or $x = 1/2$ are roots of $P(x)$.

We want to find the parameters $m$ and $n$ so that the polynomial $P(x) = 2x^4 - x^3 + mx^2 + nx + 6$ is exactly divisible by the trinomial $D(x) = 2x^2 - 3x - 2$.

Two students propose different strategies:

• Álvaro proposes carrying out traditional long division directly of $P(x)$ by $D(x)$ and setting the two coefficients of the resulting linear remainder equal to zero.

• Blanca proposes factoring $D(x) = (x - 2)(2x + 1)$ and chaining two successive Ruffini tables: first dividing by $x - 2$ setting the remainder to zero, and then dividing the resulting quotient by $2x + 1$ setting the remainder to zero again.

1. a) Compare both strategies with reasoning, evaluating the computational workload and the common risks of algebraic errors (handling fractions vs. like terms with parameters).
2. b) Execute Blanca's strategy by rigorously applying Ruffini's rule to find the exact values of $m$ and $n$, indicating the two tables and the simplified final quotient of the division $P(x) : D(x)$.
3. c) Theoretical justification: If, in a general case, a polynomial $P(x)$ is successively divided by $(x - a)$ yielding remainder $R_1 \neq 0$, and the resulting quotient is divided by $(cx - b)$ yielding remainder $R_2 \neq 0$, deduce algebraically what the expression for the overall remainder of the division of $P(x)$ by $(x - a)(cx - b)$ is, and explain why it is not simply the sum $R_1 + R_2$.