Synthetic division (Ruffini's rule)
Wilkin wants to know whether a degree-5 polynomial equals zero at $x=3$. Substituting and working with large powers is slow and easy to get wrong. Synthetic division (Ruffini's rule) does the same division as always, but working only with numbers: a row of coefficients, a value $a$, and a table with three rows. Will it also tell us whether $3$ is a root? We'll find out at the end.
How synthetic division works
Write the coefficients of $P(x)$ in a row, with a $0$ for any missing degrees, and the value $a$ off to the side. The first coefficient is brought down as is. Each following result is obtained by multiplying the last number in the bottom row by $a$ and adding the coefficient in its column.
Divide the polynomial $P(x) = 3x^3 - 2x^2 + 5x - 4$ by the binomial $x - 2$ using Ruffini's rule, and state the resulting quotient and remainder.
Divide the polynomial $P(x) = 3x^3 - 2x^2 + 5x - 4$ by the binomial $x - 2$ using Ruffini's rule, and state the resulting quotient and remainder.
Apply Ruffini's rule to divide $P(x) = x^4 - 16$ by $x + 2$. Write down clearly the quotient polynomial and the value of the remainder.
Apply Ruffini's rule to divide $P(x) = x^4 - 16$ by $x + 2$. Write down clearly the quotient polynomial and the value of the remainder.
Find the remainder of the division of the polynomial $P(x) = x^5 + 3x^4 - 2x^2 + 7$ by $x + 3$ using Ruffini's rule.
Find the remainder of the division of the polynomial $P(x) = x^5 + 3x^4 - 2x^2 + 7$ by $x + 3$ using Ruffini's rule.
Evaluating and confirming roots
The last number in the table is the remainder, and by the remainder theorem, the remainder of dividing by $x-a$ is $P(a)$. So synthetic division can be used to calculate the numerical value of a polynomial without substituting term by term, and to check whether a number is a root: if the remainder is $0$, it is.
Check, using Ruffini's rule, whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$.
Check, using Ruffini's rule, whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$.
Calculate the numerical value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 1$ for $x = -1$ by applying the Remainder Theorem using Ruffini's rule.
Calculate the numerical value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 1$ for $x = -1$ by applying the Remainder Theorem using Ruffini's rule.
Determine, without performing the full division and using the Remainder Theorem or Ruffini's rule, whether the binomial $x + 1$ is a factor of the polynomial…
Determine, without performing the full division and using the Remainder Theorem or Ruffini's rule, whether the binomial $x + 1$ is a factor of the polynomial $P(x) = x^4 + x^3 - x^2 + x + 2$.
Calculate the numerical value of the polynomial $P(x) = x^4 - 2.5x^3 + 1.5x^2 - 3x + 2$ for $x = 2$ using exclusively the setup of Ruffini's rule.
Calculate the numerical value of the polynomial $P(x) = x^4 - 2.5x^3 + 1.5x^2 - 3x + 2$ for $x = 2$ using exclusively the setup of Ruffini's rule.
Factoring with a known root
If you already know a root $a$, the table directly gives the quotient, a polynomial of one degree less. Repeat the process with that quotient whenever another root appears, until you reach degree-1 factors or a degree-2 factor with no real roots.
Factorise the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$ knowing in advance that $x = 1$ is one of its roots.
Factorise the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$ knowing in advance that $x = 1$ is one of its roots.
Fully factorise the polynomial $P(x) = 2x^3 + 3x^2 - 8x + 3$ by finding all its roots.
Fully factorise the polynomial $P(x) = 2x^3 + 3x^2 - 8x + 3$ by finding all its roots.
Fully factorise the fifth-degree polynomial $P(x) = x^5 - 5x^4 + 7x^3 - 2x^2 + 4x - 8$ into irreducible polynomials over the real numbers.
Fully factorise the fifth-degree polynomial $P(x) = x^5 - 5x^4 + 7x^3 - 2x^2 + 4x - 8$ into irreducible polynomials over the real numbers.
Real-world problems
Volumes, profits, and motion are modeled with polynomials. Finding when they equal zero, or breaking them into factors, is done with synthetic division.
The volume of a box in the shape of a right rectangular prism is given by the polynomial $V(x) = x^3 + 5x^2 + 2x - 8$. If the height of the box is known to be $x - 1$,…
The volume of a box in the shape of a right rectangular prism is given by the polynomial $V(x) = x^3 + 5x^2 + 2x - 8$. If the height of the box is known to be $x - 1$, determine the polynomial representing the base area $A(x)$.
The profits of a company, in thousands of pesos, are modeled during its first years of operation by the polynomial $B(x) = -x^3 + 10x^2 - 31x + 30$, where $x$ represents…
The profits of a company, in thousands of pesos, are modeled during its first years of operation by the polynomial $B(x) = -x^3 + 10x^2 - 31x + 30$, where $x$ represents time in years. Find the exact years in which the company's profit was zero (break-even points).
The position of a particle in straight-line motion is given by the equation $s(t) = t^3 - 7t^2 + 14t - 8$, where $t$ is the time in seconds. Determine the instants in…
The position of a particle in straight-line motion is given by the equation $s(t) = t^3 - 7t^2 + 14t - 8$, where $t$ is the time in seconds. Determine the instants in time at which the particle passes through the origin of coordinates ($s(t) = 0$).
The area of a rectangular solar panel is described by the polynomial $A(x) = 2x^3 + 7x^2 + 4x + 3$. If the width of the panel is known to be $x + 3$, find the polynomial…
The area of a rectangular solar panel is described by the polynomial $A(x) = 2x^3 + 7x^2 + 4x + 3$. If the width of the panel is known to be $x + 3$, find the polynomial representing its length.
The temperature of a chemical reaction as a function of time $t$ (in minutes) is modelled by the polynomial $T(t) = -t^4 + 8t^3 - 17t^2 + 10t$. Determine the times at…
The temperature of a chemical reaction as a function of time $t$ (in minutes) is modelled by the polynomial $T(t) = -t^4 + 8t^3 - 17t^2 + 10t$. Determine the times at which the temperature of the reaction is exactly $0^\circ\text{C}$.
The volume of a spherical cap is given by the formula $V = \frac{\pi h^2}{3}(3R - h)$, where $R$ is the radius of the sphere and $h$ is the height of the cap. If the…
The volume of a spherical cap is given by the formula $V = \frac{\pi h^2}{3}(3R - h)$, where $R$ is the radius of the sphere and $h$ is the height of the cap. If the radius of a sphere is $R = 5\text{ cm}$ and a cap with a volume of $V = \frac{52\pi}{3}\text{ cm}^3$ is required, set up the corresponding equation and use Ruffini's rule to find the height $h$.
Patterns, equations, and inequalities
Synthetic division can also be used to discover patterns. Dividing $x^3-27$ by $x-3$ with coefficients $1,0,0,-27$ gives $1$; $3$; $9$; $0$. The quotient is $x^2+3x+9$: each coefficient is a power of $3$.
To solve a polynomial equation like $x^3-7x^2+14x-8=0$, factor with the table and set each factor equal to zero. For an inequality like $P(x)\ge0$, factor, place the roots on the number line, and study the sign of the product on each interval.
Apply Ruffini's rule to divide the polynomial $P(x) = x^5 - 32$ by $x - 2$. From the pattern observed in the coefficients of the quotient, deduce and write down the…
Apply Ruffini's rule to divide the polynomial $P(x) = x^5 - 32$ by $x - 2$. From the pattern observed in the coefficients of the quotient, deduce and write down the general expression for the quotient polynomial when dividing $x^n - a^n$ by $x - a$.
Solve the polynomial inequality $x^4 - 5x^3 + 5x^2 + 5x - 6 \ge 0$. To do this, factorise the polynomial first using Ruffini's rule and then analyse the sign in the…
Solve the polynomial inequality $x^4 - 5x^3 + 5x^2 + 5x - 6 \ge 0$. To do this, factorise the polynomial first using Ruffini's rule and then analyse the sign in the corresponding intervals of the real line.
Extension: parameters from the remainder
If a coefficient is unknown, a condition like "it is divisible by $x-a$" or "the remainder is $c$" becomes an equation: the last number in the table must equal whatever the condition says.
With two parameters you set up two conditions and solve the system of two equations. If divisibility by $(x-a)^2$ is required, divide twice: both remainders must be $0$.
Determine the value of the constant $k$ so that the polynomial $P(x) = x^3 - kx^2 + 3x - 2$ is exactly divisible by $x - 1$.
Determine the value of the constant $k$ so that the polynomial $P(x) = x^3 - kx^2 + 3x - 2$ is exactly divisible by $x - 1$.
Determine the values of the parameters $m$ and $n$ so that the polynomial $P(x) = x^4 - 3x^3 + mx + n$ is exactly divisible by $x - 2$ and, at the same time, gives a…
Determine the values of the parameters $m$ and $n$ so that the polynomial $P(x) = x^4 - 3x^3 + mx + n$ is exactly divisible by $x - 2$ and, at the same time, gives a remainder of $4$ when divided by $x + 1$.
Find the values of $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 12x + 4$ is divisible by $(x - 2)^2$.
Find the values of $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 12x + 4$ is divisible by $(x - 2)^2$.
Find the expression of a cubic polynomial $P(x)$ whose leading coefficient is $1$, knowing that it equals zero for $x = 1$ and $x = 2$, and that when divided by $x - 3$…
Find the expression of a cubic polynomial $P(x)$ whose leading coefficient is $1$, knowing that it equals zero for $x = 1$ and $x = 2$, and that when divided by $x - 3$ it gives a remainder of $10$.
Extension: when the divisor is not $x-a$
Dividing by $x-\frac12$ is no problem: $a=\frac12$. But to divide by $2x-3$ you have to factor out the $2$: $2x-3=2\left(x-\frac32\right)$. You apply the table with $a=\frac32$ and the quotient from the table is divided by 2.
Divide the polynomial $P(x) = 2x^3 - 5x^2 + 3x - 1$ by the binomial $x - \frac{1}{2}$ using Ruffini's rule. Express the final result by stating the quotient and…
Divide the polynomial $P(x) = 2x^3 - 5x^2 + 3x - 1$ by the binomial $x - \frac{1}{2}$ using Ruffini's rule. Express the final result by stating the quotient and remainder.
Perform the division of the polynomial $P(x) = 4x^3 - 8x^2 + 7x - 3$ by the binomial $2x - 3$ using Ruffini's rule. Note that you must adapt the divisor to apply the…
Perform the division of the polynomial $P(x) = 4x^3 - 8x^2 + 7x - 3$ by the binomial $2x - 3$ using Ruffini's rule. Note that you must adapt the divisor to apply the algorithm and subsequently adjust the quotient and remainder obtained.