Synthetic Division and the Factor Theorem
A Leaving Cert question gives you a cubic like $2x^3+3x^2-8x+3$ and asks you to solve it or factorise it fully. Where do you even start? Dividing by $x-a$ with long division works, but it is slow and easy to get wrong. Synthetic division (a shortcut some textbooks call Ruffini's rule) does exactly the same thing using only numbers: one row of coefficients, a value $a$, and a three-row table that gives the quotient and the remainder in seconds. Combined with the Factor Theorem, it cracks cubics with an integer root. We will come back to that opening cubic at the end of the factorising section.
The mechanics of synthetic division
Write the coefficients of $P(x)$ in a row (with a $0$ for any missing degrees) and the value $a$ separately. The first coefficient is brought down unchanged; 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.
Evaluating polynomials and the Factor Theorem
The last number in the bottom row is the remainder, and the remainder of dividing by $(x-a)$ is $P(a)$. Why? Write $P(x)=(x-a)Q(x)+R$ and put $x=a$: the first term vanishes, so $P(a)=R$. So synthetic division can calculate the numerical value of a polynomial without substituting term by term, and test a root. This is the Factor Theorem: $(x-a)$ is a factor of $P(x)$ if and only if $P(a)=0$. To find integer roots, try the divisors of the constant term.
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 $P(x) = x^4 + x^3 - x^2 + x + 2$.
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.
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.
Factorising with a known root
When a root $a$ is already known, synthetic division gives the quotient directly: a factor of one degree lower. For a cubic with an integer root, the quotient is a quadratic that you factorise (or solve with the quadratic formula). Repeating the process with the quotient gives the complete factorisation.
Back to the opening cubic $2x^3+3x^2-8x+3$: try the divisors of $3$. $P(1)=2+3-8+3=0$, so $x=1$ is a root and $(x-1)$ is a factor. Dividing gives $2x^2+5x-3=(2x-1)(x+3)$.
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.
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.
Unknown parameters from a condition
When the polynomial has an undetermined coefficient, a divisibility condition ("it is divisible by $x-a$", "the remainder when dividing by $x-a$ is $c$") turns into an equation: the last number in the bottom row, which is $P(a)$, must equal whatever the condition says. With two unknowns, you get two conditions and so a $2\times2$ system of linear equations.
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 $10$.
Synthetic division in modelling problems
Volumes, profits and motion are expressed as polynomials in which it is useful to find factors: a volume that is a product of dimensions, such as a box made by a packaging company in Cork, or the times at which a quantity passes through zero.
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 euros, are modelled during its first years of operation by the polynomial $B(x) = -x^3 + 10x^2 - 31x + 30$, where $x$…
The profits of a company, in thousands of euros, are modelled 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 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 $0^\circ\text{C}$.
When the divisor is not exactly $x-a$
Synthetic division is designed for dividing by $x-a$, with coefficient $1$ on the $x$. Dividing by $x-\frac12$ is no problem: $a=\frac12$. But to divide by a binomial such as $2x-3$, you must first take out the $2$ as a factor: $2x-3=2\left(x-\frac32\right)$, apply the table with $a=\frac32$, and then divide the quotient obtained by $2$ to compensate.
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.
Cian has to divide the polynomial $P(x) = 4x^3 - 6x^2 + 8x - 5$ by the binomial $2x - 1$. To do this, he applies Ruffini's rule using the value $a = 1/2$. In the last…
Cian has to divide the polynomial $P(x) = 4x^3 - 6x^2 + 8x - 5$ by the binomial $2x - 1$. To do this, he applies Ruffini's rule using the value $a = 1/2$. In the last row of his table, he successively obtains the coefficients $4$, $-4$, $6$ and, as the last number, $-2$. From this, Cian concludes in his exam: "The quotient of the division is $C(x) = 4x^2 - 4x + 6$ and the remainder is $R = -2$". Identify and explain with reasoning the conceptual error made by Cian, prove using the division algorithm check that his quotient cannot be correct, and write the complete and corrected solution (actual quotient and remainder).
We want to find the parameters $m$ and $n$ so that the polynomial $P(x) = 2x^4 - x^3 + mx^2 + nx + 6$ is exactly divisible by the trinomial $D(x) = 2x^2 - 3x - 2$. Two…
We want to find the parameters $m$ and $n$ so that the polynomial $P(x) = 2x^4 - x^3 + mx^2 + nx + 6$ is exactly divisible by the trinomial $D(x) = 2x^2 - 3x - 2$.
Two students propose different strategies:
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Seán proposes carrying out traditional long division directly of $P(x)$ by $D(x)$ and setting the two coefficients of the resulting linear remainder equal to zero.
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Aoife proposes factoring $D(x) = (x - 2)(2x + 1)$ and chaining two successive Ruffini tables: first dividing by $x - 2$ setting the remainder to zero, and then dividing the resulting quotient by $2x + 1$ setting the remainder to zero again.
- a) Compare both strategies with reasoning, evaluating the computational workload and the common risks of algebraic errors (handling fractions vs. like terms with parameters).
- b) Execute Aoife's strategy by rigorously applying Ruffini's rule to find the exact values of $m$ and $n$, indicating the two tables and the simplified final quotient of the division $P(x) : D(x)$.
- c) Theoretical justification: If, in a general case, a polynomial $P(x)$ is successively divided by $(x - a)$ yielding remainder $R_1 \neq 0$, and the resulting quotient is divided by $(cx - b)$ yielding remainder $R_2 \neq 0$, deduce algebraically what the expression for the overall remainder of the division of $P(x)$ by $(x - a)(cx - b)$ is, and explain why it is not simply the sum $R_1 + R_2$.
Designing, testing and comparing an algorithm
Synthetic division also allows you to evaluate a polynomial. For $P(x)=a_nx^n+\cdots+a_1x+a_0$, the Horner scheme chains together multiplications and additions. First design how you would do it with pencil and paper; then compare your procedure with this pseudocode:
input: coefficients [a_n, ..., a_0] and value t
v ← a_n
for each remaining coefficient c, from highest to lowest degree:
v ← v * t + c
return v
Each arrow represents the next multiplication and addition:
For a polynomial of degree n, Horner performs n multiplications and n additions. If your direct algorithm calculates each power from scratch by repeated products and then multiplies by its coefficient, it uses $n(n+1)/2$ multiplications and n additions. If it reuses powers, the count changes: always state which algorithm you are comparing. For the cubic above, the general counts are 3 versus 6 multiplications, without simplifying zero terms.
Modify the algorithm to evaluate the same polynomial at a list of values, resetting v for each value. Write down the modification, run it by hand or in a program, and keep a table of input, expected output and actual output. Also check $t=1$: the result must be the sum of the coefficients. Explain why, after processing every coefficient, v represents exactly the nested polynomial $((2t-3)t+0)t+5$.
To calculate the numerical value of the polynomial $P(x) = 2x^4 - 5x^3 + 3x - 7$ at $x = 3$, two different approaches can be considered: substituting $x = 3$ directly…
To calculate the numerical value of the polynomial $P(x) = 2x^4 - 5x^3 + 3x - 7$ at $x = 3$, two different approaches can be considered: substituting $x = 3$ directly and calculating each power, or applying Ruffini's rule by dividing by $x - 3$ to find the remainder.
- a) Compare both methods by analyzing the type and number of operations that each one requires (high powers versus successive additions and multiplications) and justify which one is faster and less error-prone when a calculator is not available.
- b) Apply Ruffini's rule (paying attention to missing terms) to obtain the remainder, deduce the value of $P(3)$, and state with reasoning whether $x - 3$ is a factor of $P(x)$.