The Remainder Theorem and the Factor Theorem

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

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.

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

Determine if the polynomial $Q(x) = x^3 + 4x^2 - x - 4$ is divisible by $x + 4$ by applying the remainder theorem.

Check using the remainder theorem if $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + 5x - 6$.

The growth of a colony of bacteria in a controlled medium is modeled 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.

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 $4$.

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.

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 $a$.

The position of a moving body 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 feet). Use the remainder theorem to calculate the position of the moving body at the instant $t = 3$ and explain the conceptual relationship between the value found and the polynomial division of $x(t)$ by $t - 3$.

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

The deflection of a beam at a point $x$ (in feet) is given by the polynomial $D(x) = kx^4 - 2x^3 + 3x^2$. Knowing that the deflection at $x = 2$ is $20$ inches, find the value of the constant $k$ relying on the remainder theorem and determine what the deflection will be at the coordinate $x = -1$.

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 $x + 2$.

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 $(x + 1)$.

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 $x + 1$.

Let the polynomial be $P(x) = x^3 + ax^2 + bx - 4$. 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 $b$.

The heat capacity of an experimental material is approximated by the third-degree polynomial $C(T) = aT^3 + bT^2 + cT + d$. We know 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 analytical expression of the polynomial $C(T)$.

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

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

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 $x = 1$.

In a cryptographic security system, a key is generated by evaluating a polynomial of a 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 logically deduce the remainder of the division of $x^5 - 32$ by $x - 2$.

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

Note: Remember that the remainder of 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.

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 $-12$.