Remainder Theorem
Yaritza works at a shop in Santiago and wants to know how much is left over when she divides a cost polynomial by $x-3$. Doing the whole division is long and easy to get wrong. There is a shortcut: with a single substitution, the problem is solved in seconds. That shortcut is the Remainder Theorem, the basis of all the analysis of the roots of a polynomial equation.
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 $x - 2$.
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…
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 $x + 1$.
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.
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.
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 theorem applied to the case where the remainder vanishes.
Graphically, the real roots are the points where the curve crosses the $x$-axis.
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 a condition
When a coefficient is unknown and the remainder is given (or the division is required to be exact), you substitute in the same way and the condition becomes an equation for that unknown. You can also compare two remainders, as in "the remainder of $P$ divided by $x-2$ equals that of $Q$ divided by $x+1$": you evaluate each one and set them equal.
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 $a$.
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…
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 $x + 1$.
Building a polynomial from its roots
If you know that $r_1$, $r_2$ and $r_3$ are roots of a third-degree polynomial, the Factor Theorem says that $(x-r_1)$, $(x-r_2)$ and $(x-r_3)$ are factors. Since the degree is $3$, there is room for no more factors than those, apart from a number that multiplies everything: the leading coefficient.
If one of the roots is not known, it is left as an unknown and found with another condition, for example a given value $P(b)$.
A useful trick: if a polynomial leaves the same remainder $c$ when divided by several binomials $(x-a)$, then $P(a)=c$ for all those values, and therefore $P(x)-c$ vanishes at them: $P(x)-c$ has those factors.
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.
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.
Applications: model values and roots
Many models (position, cost, growth) are polynomials. Evaluating $P(a)$ gives the value of the model at $a$, and the Remainder Theorem interprets it: it is the remainder of dividing by $x-a$. If $P(a)=0$, the model equals zero at that point and $a$ is a root.
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 meters). 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 meters). 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 production cost of $q$ thousand units of an electronic component follows the formula $C(q) = 30q^4 - 180q^3 + 300q^2 + 720$ (in thousands of pesos). Apply the…
The production cost of $q$ thousand units of an electronic component follows the formula $C(q) = 30q^4 - 180q^3 + 300q^2 + 720$ (in thousands of pesos). Apply the Remainder Theorem to determine the exact cost of manufacturing $4000$ units ($q=4$).
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…
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.
The deflection of a beam at a point $x$ (in meters) 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…
The deflection of a beam at a point $x$ (in meters) 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 $x = -1$.
Extension: huge exponents
The theorem does not distinguish between a polynomial of degree $3$ and one of degree $2024$: substituting costs the same. You only have to be careful with the powers, and sometimes terms cancel.
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 $x - 2$.
Extension: remainder when dividing by a degree-2 polynomial
If the divisor has degree $2$, the remainder can have degree $1$: $R(x)=mx+n$. If you know the two roots $r_1$ and $r_2$ of the divisor, evaluate at each one.
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.
Back to the beginning: Yaritza does not need to divide. It is enough to evaluate the polynomial at $x=3$.