Remainder theorem
To find the remainder when $P(x)=x^{2024}-3x^{2023}+2x^5-7x+4$ is divided by $x-3$, nobody carries out the whole division: with a single substitution, $P(3)$, the problem is solved in seconds. The remainder theorem is that substitution, and in Bachillerato it is used with more confidence: several conditions at once, enormous indices, or divisors of degree $2$.
The remainder theorem
Without performing the division, calculate the remainder when dividing the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$.
Without performing the division, calculate the remainder when dividing the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$.
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…
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 $t - 3$.
The factor theorem: when the remainder is zero
If the remainder $P(a)$ turns out to be $0$, the division is exact: $(x-a)$ is a factor of $P(x)$ and $a$ is a root. It is the same idea as the remainder theorem, applied to the particular case in which the remainder vanishes.
Determine whether the polynomial $Q(x) = x^3 + 4x^2 - x - 4$ is divisible by $x + 4$ by applying the Remainder Theorem.
Determine whether 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 whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + 5x - 6$.
Check using the Remainder Theorem whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + 5x - 6$.
Finding a parameter from one remainder condition
When the polynomial has an unknown coefficient and a particular remainder is known (or required), we substitute in the same way, and the condition becomes an equation in that unknown.
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 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.
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 $a$.
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 $a$.
Several conditions at once: systems of equations
When there are two unknown parameters, two conditions are needed (two remainders, or one remainder and one divisibility condition), and each gives an equation. Together they form a system that is solved like any other.
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…
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 $(x + 1)$.
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 $(x + 1)$.
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…
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.
Evaluating the "unevaluable": enormous indices
The remainder theorem makes no distinction between a polynomial of degree $3$ and one of degree $2024$: substituting $P(a)$ costs just as little effort, although the powers of $a$ have to be handled carefully (sometimes it is worth factorising before substituting, to avoid enormous calculations).
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…
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…
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 remainder when dividing by a polynomial of degree 2
The remainder theorem, in its basic version, only gives the remainder when dividing by $x-a$ (degree $1$), which is a number. If the divisor has degree $2$, the remainder can be a polynomial of degree $1$, of the form $mx+n$. If the two roots of the divisor are known, evaluating $P(x)$ at each of them gives two equations from which to find $m$ and $n$.
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…
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.
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…
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.