Exercises: Remainder Theorem

Without performing the division, calculate the remainder when dividing the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$.

Determine whether the polynomial $Q(x) = x^3 + 4x^2 - x - 4$ is divisible by $x + 4$ by applying the Remainder Theorem.

Find the value of the parameter $k$ so that the remainder of dividing $P(x) = 2x^3 - kx^2 + 3x + 1$ by $x - 1$ is $4$.

Calculate the numerical value of the polynomial $P(x) = -x^5 + 3x^3 - 2x^2 + x - 5$ for $x = -1$ and state the meaning of this value with respect to the polynomial division of $P(x)$ by $x + 1$.

Find the value of $m$ knowing that the division of the polynomial $R(x) = x^4 + mx^3 - 2x^2 + 5x - 6$ by $x - 2$ is exact.

Determine the remainder of the division of $P(x) = \frac{1}{2}x^3 - \frac{3}{2}x^2 + x - \frac{1}{2}$ by $x - 2$ by evaluating the polynomial.

Check using the Remainder Theorem whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + 5x - 6$.

Knowing that the remainder of dividing $P(x) = x^3 - 2x^2 + ax + 5$ by $x + 2$ is $-7$, calculate the value of the coefficient $a$.

Lucas tries to find out whether the binomial $x + 2$ is a factor of the polynomial $P(x) = 2x^3 + 5x^2 - x + k$. To solve it, he writes in his notebook: "By the remainder theorem, I calculate $P(2) = 2 \times 2^3 + 5 \times 2^2 - 2 + k = 16 + 20 - 2 + k = 34 + k$. For it to be a factor, the remainder must be zero, so $34 + k = 0 \implies k = -34$". Rigorously identify and explain the conceptual error made by Lucas, clarify what numerical value he should have actually calculated according to the remainder theorem, and find the correct value of the parameter $k$ for the division to be exact.

Given the polynomial $P(x) = 3x^4 - 20x^3 + 5x - 12$:

1. a) Before performing any detailed calculation on paper, mentally estimate the sign of the remainder when dividing $P(x)$ by $x - 4$. To do this, analyze the two highest-degree terms when substituting $x = 4$ and reason whether the positive or negative contribution will dominate.
2. b) Apply the remainder theorem to calculate the exact value of this remainder. Check whether your qualitative estimate was correct and justify why the leading term ($3x^4$) failed to make the final value positive.

The position of an object moving in a straight line as a function of time $t$ (in seconds) is given by the polynomial $x(t) = -2t^3 + 8t^2 - 5t + 3$ (in metres). Use the Remainder Theorem to calculate the position of the object at time $t = 3$ and explain the conceptual relationship between the value found and the polynomial division of $x(t)$ by $t - 3$.

Find the values of the parameters $m$ and $n$ so that the polynomial $P(x) = x^3 + mx^2 + nx - 10$ is divisible by $x - 2$ and leaves a remainder of $-24$ when divided by $x + 2$.

The production cost of $q$ thousand units of an electronic component follows the formula $C(q) = 0.5q^4 - 3q^3 + 5q^2 + 12$ (in thousands of euros). Apply the Remainder Theorem to determine the exact cost of manufacturing $4000$ units ($q=4$).

Determine the value of $k$ so that the remainder of dividing the polynomial $P(x) = 2x^3 - kx^2 + x - k$ by $x - 2$ is identical to the remainder of dividing the polynomial $Q(x) = x^2 - kx + 3$ by $x + 1$.

The growth of a bacterial colony in a controlled environment is modelled by the polynomial $P(t) = t^4 - 7t^3 + 17t^2 - 17t + 6$, where $t$ is the time in days and $P(t)$ represents the deviation from the initial population. Prove, using the Factor Theorem, that on days $t=1$, $t=2$, and $t=3$ the population returns exactly to its initial level.

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

The deflection of a beam at a point $x$ (in metres) is given by the polynomial $D(x) = kx^4 - 2x^3 + 3x^2$. Knowing that the deflection at $x = 2$ is $20$ mm, find the value of the constant $k$ using the Remainder Theorem and determine what the deflection will be at coordinate $x = -1$.

A cubic polynomial $P(x)$ has leading coefficient $2$. Knowing that it is divisible by $(x - 1)$ and by $(x + 2)$, and that when divided by $(x - 2)$ it leaves a remainder of $16$, determine the full expression of the polynomial $P(x)$ in expanded form.

To determine whether the polynomial $P(x) = x^5 - 3x^4 + 2x^3 - x^2 + 5x - 4$ is divisible by the binomial $x - 1$, three Leaving Certificate students propose different strategies:

• Aoife proposes performing Euclidean polynomial long division to see if the final remainder is the zero polynomial.

• Seán proposes applying Ruffini's rule by completing the table of coefficients and checking if the last cell is zero.

• Niamh proposes applying the factor theorem by directly evaluating $P(1)$ through arithmetic calculation.

1. a) Reasonably compare the three methods considering the number of operations, the probability of making sign errors, and the resolution time.
2. b) Choose the most efficient method according to your analysis and apply it to determine with justification whether $x - 1$ is a factor of $P(x)$.

Met Éireann, the Irish Meteorological Service, records the evolution of temperature $T$ (in $^{\circ}\text{C}$) at a station in the Wicklow Mountains during a winter day using the polynomial function $T(t) = -0.05t^3 + 0.6t^2 - 1.1t - 2$, where $t$ represents the time in hours elapsed since 08:00 ($t = 0$), with $t \in [0, 12]$.

1. a) Interpret the physical meaning of the remainder of dividing $T(t)$ by the binomial $t - 6$ and calculate it using the remainder theorem to find the exact temperature at 14:00.
2. b) The winter road maintenance team needs to know at what time the temperature first crossed exactly the freezing point ($0\ ^{\circ}\text{C}$). Use the factor theorem to check whether $t - 4$ is an exact divisor of $T(t)$ and interpret the result obtained in the context of thawing on the road.

Determine the coefficients $a, b, c, d$ of the polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$ knowing that it is divisible by the binomial $x^2 - 4$, that when divided by $x - 1$ it leaves a remainder of $15$, and that its constant term is $-12$.

In a cryptographic security system, a key is generated by evaluating a polynomial of very high degree. The key corresponds exactly to the remainder of dividing the polynomial $P(x) = x^{2024} - 3x^{2023} + 2x^5 - 7x + 4$ by the binomial $x - 3$. Calculate the numerical value of this security key, justifying your answer.

Prove, applying the Remainder Theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n$. Use this result to deduce, with reasoning, the remainder of the division of $x^5 - 32$ by $x - 2$.

The heat capacity of an experimental material is approximated by the cubic polynomial $C(T) = aT^3 + bT^2 + cT + d$. It is known that $C(T)$ is divisible by $T$, that at $T=1$ its value is $4$, at $T=-1$ its value is $0$, and at $T=2$ its value is $18$. Formulate the corresponding system of equations using the Remainder Theorem and find the algebraic expression for the polynomial $C(T)$.

Find the remainder when dividing the polynomial $P(x) = (x - 2)^{200} + (x - 1)^{100} - 5$ by the quadratic polynomial $Q(x) = (x - 1)(x - 2)$.

Note: Remember that the remainder when dividing by a polynomial of degree 2 is at most of the form $mx + n$. Apply the Remainder Theorem by strategically evaluating at the roots of the divisor.

The profile of a drainage channel in an architectural project follows the equation $y = P(x) = -kx^3 + 2kx^2 - 5x + c$, where $y = 0$ represents ground level. It is known that the profile crosses ground level at the points $x = 5$ and $x = -2$. Find the exact values of $k$ and $c$ using the Factor Theorem, and determine the elevation of the profile (height relative to the ground, negative if below it) at the point $x = 1$.

Let $P(x) = x^3 + ax^2 + bx - 4$ be a polynomial. Knowing that it is divisible by $x - 2$ and that the sum of the remainders of dividing it by $x - 1$ and by $x + 1$ is equal to $-10$, set up the relevant system of equations and determine the coefficients $a$ and $b$.

Find the expanded expression of a cubic polynomial $P(x)$ such that, when divided by $x+1$, $x-2$, and $x+3$, it leaves the same remainder equal to $5$ in all cases, and when divided by $x-1$ it leaves a remainder of $-7$.

Note: Think about the relationship between the auxiliary polynomial $P(x) - 5$ and the Factor Theorem.

A polynomial $P(x)$ with real coefficients, whose complete expression and degree are unknown, satisfies two conditions simultaneously: when divided by $x - 2$ it yields a remainder equal to $7$, and when divided by $x + 3$ it yields a remainder equal to $-8$.

1. a) Without knowing the quotient of the division, reasonably deduce the algebraic expression of the remainder $R(x)$ obtained by dividing $P(x)$ by the second-degree polynomial $D(x) = x^2 + x - 6$, justifying the maximum degree that this remainder can have.
2. b) Design and write the expression of a specific monic polynomial of degree $3$ that satisfies both properties simultaneously and algebraically verify that dividing it by $x^2 + x - 6$ reproduces the remainder $R(x)$ deduced in the previous section.

Consider the family of high-degree polynomials $P(x) = x^{2026} + ax^{2025} + b$, where $a$ and $b$ are unknown real constants.

1. a) Use the remainder and factor theorems to set up and solve a linear system that determines the exact values of $a$ and $b$ knowing that $P(x)$ is divisible by $x + 1$, and that when divided by $x - 1$ a remainder of $4$ is obtained.
2. b) With the values obtained in the previous section, formally justify whether or not the resulting polynomial is divisible by the quadratic binomial $x^2 - 1$.
3. c) Reason analytically, without using derivatives or graphs, why this specific polynomial cannot have any strictly positive real root, analyzing the structure and the sign of its terms for any $x > 0$.