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 $P(x) = x^4 + x^3 - x^2 + x + 2$.

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 pesos, are modeled 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.

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.