Exercises: Remainder theorem

Find the remainder of the division of the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$ without performing the division.

Knowing that the polynomial is $Q(x) = x^3 + 4x^2 - x + 10$, calculate the numerical value $Q(-3)$ and explain how this value is related to the division of $Q(x)$ by $x + 3$.

Determine, using the remainder theorem, whether the polynomial $R(x) = 2x^3 - 5x^2 + x + 2$ is divisible by $x - 2$.

Calculate the remainder of dividing the polynomial $S(x) = -x^5 + 3x^2 - 4$ by $x + 1$ by applying the remainder theorem.

Check using the remainder theorem whether $x = -2$ is a root of the polynomial $T(x) = x^4 + 3x^3 - 2x^2 + 12$.

Calculate the remainder of dividing $U(x) = \frac{1}{2}x^3 - x^2 + 3x - \frac{5}{2}$ by $x - 2$ without dividing using Ruffini's rule.

Knowing that the remainder of dividing a polynomial $P(x)$ by $x - 4$ is $15$, state the numerical value of $P(4)$ and justify your answer using the remainder theorem.

Find the remainder of dividing $V(x) = 4x^6 - 3x^4 + 2x^2 - 1$ by $x + 1$ by evaluating the polynomial at the appropriate value.

Lucía has to find the remainder of dividing the polynomial $P(x) = 2x^3 - 5x + 7$ by $(x + 2)$ without carrying out long division or using Ruffini's rule. In her notebook, she writes:

"I apply the Remainder Theorem by directly substituting the number that accompanies $x$:

$R = P(2) = 2 \cdot 2^3 - 5 \cdot 2 + 7 = 2 \cdot 8 - 10 + 7 = 16 - 10 + 7 = 13$.

Since the remainder is $13 \neq 0$, the division is not exact and $(x + 2)$ is not a factor of $P(x)$".

1. a) Identify the conceptual error Lucía made and explain, using the division algorithm identity $P(x) = (x - a) \cdot C(x) + R$, which value she should have actually substituted into the polynomial and why.
2. b) Correct the exercise by calculating the true remainder and state whether the conclusion about whether $(x + 2)$ is a factor of $P(x)$ holds or changes.

A delivery van from a courier company in Zaragoza is carrying out its morning route. The remaining energy in its battery (measured in kilowatt-hours, $\text{kWh}$) after $x$ hours of continuous driving is modelled by the polynomial $B(x) = -x^3 + 3x^2 - 16x + 48$, with $0 \le x \le 3$.

1. a) Justify, using the Remainder Theorem, why the remainder of dividing $B(x)$ by $(x - 3)$ indicates exactly how much energy the van has left at the end of the third hour of delivery.
2. b) Apply the Remainder Theorem to calculate this value without performing polynomial division and interpret the result in practice: has the battery completely run out or can it continue driving? Is $(x - 3)$ a factor of the polynomial $B(x)$?

Calculate the value of the parameter $k$ so that the remainder of dividing the polynomial $P(x) = x^3 - kx^2 + 5x - 2$ by $x - 3$ is equal to $10$.

The trajectory of a racing drone on a test run is given by the polynomial $A(t) = 2t^3 - mt^2 + 4t + 5$, where $t$ is the time in seconds and $A(t)$ is the height in metres. If at $2$ seconds the height is $17$ metres, use the remainder theorem to determine the value of the parameter $m$.

Find the value of $a$ so that the polynomial $Q(x) = 2x^4 + ax^3 - 3x^2 + 5x - a$ is divisible by $x + 2$.

Determine the value of $b$ knowing that dividing the polynomial $R(x) = 3x^3 + bx^2 - 2x + 4$ by $x - 1$ gives the same remainder as dividing it by $x + 2$.

Calculate the remainder of dividing the polynomial $S(x) = x^{100} - 3x^{99} + 5$ by $x - 1$ by applying the remainder theorem.

In a chemistry experiment, the concentration of a reactant over time is modelled using the polynomial $C(t) = t^4 - 4t^3 + kt + 8$. Knowing that $C(t)$ is divisible by $t - 2$, find the value of $k$ and then calculate the remainder of dividing $C(t)$ by $t - 3$.

Find the remainder of dividing the polynomial $P(x) = 4x^3 - 6x^2 + 2x - 1$ by the binomial $2x - 1$. Take into account the value that makes the divisor zero to apply the remainder theorem.

The daily revenue of a shop, in hundreds of euros, is approximated by the polynomial $I(x) = -x^3 + 12x^2 - 20x + c$, where $x$ is the number of days elapsed. If the remainder of dividing $I(x)$ by $x - 5$ is $80$, find the value of the constant term $c$ using the remainder theorem.

To verify the stability of a data compression algorithm, a programming team needs to determine whether the polynomial $P(x) = x^{24} - 5x^8 + 4$ is exactly divisible by $(x - 1)$ and by $(x + 1)$. Two team members discuss the procedure:

• Sergio states: "The safest approach is to apply Ruffini's rule in both cases, because it is the standard method for dividing by binomials of the form $(x - a)$ and it also gives us the quotient".

• Marta replies: "In this specific case, applying Ruffini's rule is an unnecessary waste of time, and it is much more reasonable to use the Remainder Theorem".

1. a) Analyse the practical difficulties and the risk of error that applying Ruffini's rule to this polynomial would entail (mention how many coefficients would have to be written and their value).
2. b) Justify which of the two viewpoints is appropriate considering the information requested by the problem.
3. c) Use the most efficient method to calculate the remainders of both divisions and conclude with justification whether $(x - 1)$ and $(x + 1)$ are factors of $P(x)$.

The State Meteorological Agency (AEMET) models the temperature $T(t)$ (in $^{\circ}\text{C}$) at a weather station in Molina de Aragón (Guadalajara) during a January night using the polynomial:

$T(t) = 0,5t^3 - 3t^2 + 2t + 4$

where $t$ represents the hours elapsed since midnight ($0 \le t \le 6$). The meteorological team wants to know the remainder of dividing $T(t)$ by $(t - 4)$, which coincides with the exact temperature at $4:00\text{ h}$ in the morning.

1. a) Estimate: Considering that at $0:00\text{ h}$ ($t = 0$) the temperature was $4\ ^{\circ}\text{C}$ and that severe frosts occur during clear winter nights in that area, make a reasoned prior estimate: do you expect the remainder of the division to be positive or negative? Do you think the division will be exact ($R = 0$)?
2. b) Verify: Calculate the remainder of dividing $T(t)$ by $(t - 4)$ applying the Remainder Theorem.
3. c) Compare the obtained value with your estimate, state whether $(t - 4)$ is a factor of $T(t)$, and interpret what the numerical result implies regarding the risk of water freezing on the ground ($0\ ^{\circ}\text{C}$).

A quadratic polynomial $P(x)$ satisfies that when divided by $x - 1$ the remainder is $4$, when divided by $x + 2$ the remainder is $19$, and when divided by $x$ the remainder is $1$. Find the complete algebraic expression for $P(x)$.

When a polynomial $P(x)$ is divided by $x - 2$, the remainder is $5$. When the same polynomial is divided by $x + 3$, the remainder is $-5$. Calculate the remainder of dividing $P(x)$ by the product $(x - 2)(x + 3)$, knowing that this remainder must be a polynomial of degree less than $2$.

The design of a sports car bodywork profile is mathematically modelled using a polynomial $D(x) = x^4 + ax^3 + bx^2 - 4x + 4$. To ensure optimal aerodynamics, the polynomial must be divisible by $x - 2$ and leave a remainder of $15$ when divided by $x + 1$. Set up a system of equations using the remainder theorem and calculate the values of $a$ and $b$.

Determine the value of the parameters $m$ and $n$ so that the polynomial $P(x) = x^3 + mx^2 + nx - 6$ is divisible by $x^2 - 5x + 6$. Apply the remainder theorem by evaluating at the roots of the divisor.

Calculate the remainder of the division of the polynomial $P(x) = (x - 3)^{2024} + (x - 2)^{2023} - x + 5$ by $x - 3$ by applying the remainder theorem.

Prove, using the remainder theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \geq 1$.

In a technology company, the production cost function of a new microchip is modelled with a cubic polynomial $C(x) = ax^3 + bx^2 + cx + d$. It is known that $C(0) = 10$, and that the remainders when $C(x)$ is divided by $x - 1$, $x + 1$ and $x - 2$ are $12$, $6$ and $30$ respectively. Calculate the coefficients $a, b, c, d$.

Find the value of the parameter $k$ so that the remainder when dividing $P(x) = x^3 - 2kx^2 + k^2x - 8$ by $x - 2$ is exactly twice the remainder when dividing $P(x)$ by $x - 1$.

Find the analytic expression of a second-degree polynomial, $P(x) = ax^2 + bx + c$, that simultaneously satisfies these three conditions:

1. Its graph intersects the vertical axis at the point $(0, -6)$ (that is, the remainder when dividing $P(x)$ by $x$ is $-6$).

2. When dividing $P(x)$ by $(x - 3)$, the division is exact.

3. When dividing $P(x)$ by $(x + 1)$, the remainder obtained is precisely the opposite (same absolute value but opposite sign) of the remainder resulting from dividing it by $(x - 2)$.

1. a) Translate each of the conditions into algebraic equations using the Remainder Theorem.
2. b) Solve the resulting system to calculate the coefficients $a$, $b$, and $c$ precisely, and write the complete formula for $P(x)$.
3. c) Numerically verify that the obtained polynomial satisfies condition 2 by demonstrating that $x = 3$ is a root of $P(x)$.

A study on ecological transition prepared with data from the National Statistics Institute (INE) models the recycling rate of plastic waste in an autonomous community using the polynomial:

$R(t) = -t^3 + 9t^2 + k \cdot t + 30$

where $t$ is the time in years elapsed since 2020 ($t = 0$ corresponds to 2020, $t = 1$ to 2021, etc.), $R(t)$ represents the recycling percentage achieved, and $k$ is a parameter that depends on investment in awareness campaigns.

1. a) European regulations required a recycling rate of $75\,\%$ for the year 2023 ($t = 3$). Knowing that, according to the Remainder Theorem, the division of $R(t)$ by $(t - 3)$ must give exactly $75$ as a remainder, calculate the value of parameter $k$.
2. b) With the value of $k$ obtained in the previous section, apply the Remainder Theorem to find the remainder of dividing $R(t)$ by $(t - 5)$ and explain what real magnitude this number represents in the context of the year 2025.
3. c) Critically assess the result obtained in section b): is this polynomial model acceptable for making forecasts in the year 2025 and beyond? Justify your answer taking into account the definition and logical limits of a percentage rate.