Ruffini's Rule (Synthetic Division)
A packaging factory in Puebla knows that the volume of its boxes is $V(x)=x^3+5x^2+2x-8$ and that the height is $x-1$. How can you quickly find the area of the base without doing the whole long division? Synthetic division (Ruffini's rule) does the same thing as long division, but working only with the coefficients. At the end of the page we solve this box.
The mechanics of synthetic division
Write the coefficients of $P(x)$ in a row (with a $0$ for every missing degree) and, off to the side, the number $a$ from the divisor $x-a$. Bring down the first coefficient. Each following number in the bottom row is obtained by multiplying the previous one 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 polynomials 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 $P(x) = x^4 + x^3 - x^2 + x + 2$.
Fractional and decimal coefficients, and divisors $cx-a$
The table works the same way with fractions or decimals: to divide by $x-\frac12$ you use $a=\frac12$, and to evaluate at $x=2$ a polynomial with coefficients such as $-2.5$ or $1.5$ you do the same multiplications and additions.
But to divide by a binomial like $2x-3$, the coefficient of $x$ is not $1$. We factor out the common factor: $2x-3=2\left(x-\frac32\right)$, apply the table with $a=\frac32$, and then divide the quotient 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.
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.
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.
Finding roots and factoring
If you already know a root $a$, the table gives you the quotient, a factor with degree one less. You repeat with that quotient until you reach a quadratic, which you factor or solve with the quadratic formula.
And if you aren't given the root? If $P(x)$ has integer coefficients, the rational roots $\frac{p}{q}$ satisfy that $p$ divides the constant term and $q$ divides the leading coefficient. You test those candidates with the table.
If the final quotient is a quadratic with a negative discriminant, it cannot be factored over the real numbers: it is an irreducible factor. For example, $x^3-x^2+x-1=(x-1)(x^2+1)$, since the table with $a=1$ gives $1,0,1,0$ and $x^2+1$ has 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.
Parameters from a condition
When the polynomial has an unknown coefficient, the condition in the problem becomes an equation: the remainder from the table must equal whatever the statement says. If it is "exactly divisible," the remainder is $0$. If it says "leaves remainder $5$," the remainder is $5$.
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 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 $10$.
Repeated roots: divisible by $(x-a)^2$
A polynomial is divisible by $(x-a)^2$ if $a$ is a root twice: you apply the table with $a$, the remainder must be $0$, and then you apply the table with $a$ again to the quotient, where the remainder must also be $0$. Each remainder gives you an equation.
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 $(x - 2)^2$.
A pattern: $x^n-a^n$ divided by $x-a$
Divide $x^3-8$ by $x-2$: the row is $1,0,0,-8$ with $a=2$ and the table gives $1$, $2$, $4$, $0$. The quotient is $x^2+2x+4$ and the remainder is $0$ (of course: $P(2)=0$).
Each number in the quotient is the previous one multiplied by $a$, because the zeros add nothing. So the coefficients are powers of $a$.
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 $x - a$.
Problems with volumes, profits, and motion
Volumes, profits, and positions are expressed with polynomials. If the volume is the product of three dimensions and you know one, synthetic division gives you the product of the other two. If they ask when something equals zero (zero profit, position at the origin), you look for the roots of the polynomial.
Back to the beginning: the base of the box has area $x^2+6x+8$, and we also know that its sides measure $x+2$ and $x+4$.
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 ($s(t) = 0$).
Polynomial equations and inequalities
To solve $P(x)=0$ of degree $3$ or higher, you factor with the table and set each factor equal to zero. First take out a common factor if there is one: in a polynomial with no constant term, $x$ is a factor. When a problem gives you a formula (for example, a volume), turn it into the form $P(x)=0$ before looking for roots.
For an inequality $P(x)\ge0$, you factor, place the roots on the real number line, and check the sign of $P$ in each interval.
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 $h$.
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 $0^\circ\text{C}$.
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.