The Remainder Theorem
Imagine that your training app calculates how many calories you have burned using a polynomial formula, and you want to know the result after exactly two hours, without having to rebuild the whole table of earlier values. The remainder theorem does just that with polynomials: it gives you the remainder of a division without dividing, using a single substitution.
The 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.
Find the remainder of the division of the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$ without performing the division.
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.
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.
Remainders with $x + a$: watch the sign
Dividing by $x + a$ is the same as dividing by $x - (-a)$, so the value you substitute is not $a$, but $-a$.
Calculate the remainder of dividing the polynomial $S(x) = -x^5 + 3x^2 - 4$ by $x + 1$ by applying the remainder theorem.
Calculate the remainder of dividing the polynomial $S(x) = -x^5 + 3x^2 - 4$ by $x + 1$ by applying 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.
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.
Zero remainder: the link with roots
The most important case of the theorem is when the remainder is exactly $0$.
Determine, using the remainder theorem, whether the polynomial $R(x) = 2x^3 - 5x^2 + x + 2$ is divisible by $x - 2$.
Determine, using the remainder theorem, whether the polynomial $R(x) = 2x^3 - 5x^2 + x + 2$ is divisible by $x - 2$.
Check using the remainder theorem whether $x = -2$ is a root of the polynomial $T(x) = x^4 + 3x^3 - 2x^2 + 12$.
Check using the remainder theorem whether $x = -2$ is a root of the polynomial $T(x) = x^4 + 3x^3 - 2x^2 + 12$.
Finding a parameter for a given remainder
When a polynomial has an unknown letter (a parameter) and you are told what remainder it must give when divided by $(x-a)$, the theorem turns the problem into a simple equation: substitute, set the result equal to the required remainder and solve.
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 $10$.
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 $10$.
Find the value of $a$ so that the polynomial $Q(x) = 2x^4 + ax^3 - 3x^2 + 5x - a$ is divisible by $x + 2$.
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 $x + 2$.
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 $x + 2$.
Several conditions at once
Some problems give two or more conditions about remainders, with two unknown parameters: each condition becomes an equation, and you solve the system.
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…
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.
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…
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 $b$.
Back to the start: To find out how many calories you have burned after exactly two hours without rebuilding the table, you just need to calculate the numerical value by substituting $x = 2$ into the app's polynomial formula.