Factorisation and Ruffini's rule
A factorised polynomial is, for a program that solves equations, what an exploded-view plan is for someone assembling a piece of furniture: instead of one compact piece, it directly gives the parts along which it can be split. Factorising $P(x)=2x(x+5)(x-4)^2$ tells you at a glance that $P(x)=0$ exactly when $x=0$, $x=-5$ or $x=4$. Ruffini's rule is the tool that turns any polynomial into that exploded-view plan.
Common factor and special identities
Factorising always starts the same way: take out all the possible common factor (the HCF of the coefficients and, for each letter, the lowest power with which it appears in all the terms) and, if what is left is a special identity, factorise that too.
Factorise out the common factor in the following polynomial, simplifying the expression as much as possible: $P(x) = 15x^5 - 10x^4 + 20x^3$
Factorise out the common factor in the following polynomial, simplifying the expression as much as possible:
$P(x) = 15x^5 - 10x^4 + 20x^3$
Factorise the following polynomials by recognising algebraic identities (square of a sum, square of a difference, or difference of two squares): a)…
Factorise the following polynomials by recognising algebraic identities (square of a sum, square of a difference, or difference of two squares):
- a) $A(x) = x^2 - 14x + 49$
- b) $B(x) = 9x^2 - 16$
Ruffini's rule for factorising
To factorise a polynomial of degree $\geq 3$, the common factor and the identities are usually not enough. The central tool is Ruffini's rule: if a root $a$ is known (or suspected), Ruffini's rule divides $P(x)$ by $(x-a)$ and directly gives the quotient, a factor whose degree is one less.
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. Clearly state the quotient polynomial and the remainder of the…
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. Clearly state the quotient polynomial and the remainder of the division.
Given that $x = 1$ is one of the roots of the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$, use Ruffini's rule to factorise it and find the other two roots.
Given that $x = 1$ is one of the roots of the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$, use Ruffini's rule to factorise it and find the other two roots.
Divide the polynomial $P(x) = 4x^3 - 8x^2 + 5x - 1$ by the binomial $(2x - 3)$. Hint: You can use Ruffini's rule by dividing by $(x - \frac{3}{2})$ and then adjusting…
Divide the polynomial $P(x) = 4x^3 - 8x^2 + 5x - 1$ by the binomial $(2x - 3)$. Hint: You can use Ruffini's rule by dividing by $(x - \frac{3}{2})$ and then adjusting the quotient obtained.
The remainder theorem and the factor theorem
Before diving into Ruffini's rule with any candidate, it is worth checking quickly whether it is worthwhile: that is what the remainder theorem is for.
Without performing the division, use the remainder theorem to calculate the remainder obtained when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.
Without performing the division, use the remainder theorem to calculate the remainder obtained when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.
Use the factor theorem to check whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your calculations.
Use the factor theorem to check whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your calculations.
Find the value of the parameter $k$ so that the polynomial $P(x) = x^3 + kx^2 - 3x + 8$ gives a remainder of $10$ when divided by $(x - 2)$.
Find the value of the parameter $k$ so that the polynomial $P(x) = x^3 + kx^2 - 3x + 8$ gives a remainder of $10$ when divided by $(x - 2)$.
Candidates for integer roots
Trying numbers at random is inefficient. If $P(x)$ has integer coefficients, its integer roots (if it has any) are always among the divisors of the constant term.
It is known that the polynomial $P(x) = 2x^3 + x^2 - 13x + 6$ has at least one integer root. Use the remainder theorem or Ruffini's rule to check that $x = 2$ is one of…
It is known that the polynomial $P(x) = 2x^3 + x^2 - 13x + 6$ has at least one integer root. Use the remainder theorem or Ruffini's rule to check that $x = 2$ is one of its roots. Next, fully factorise the polynomial to find its other two roots (one of which is a fraction).
A team of engineers analyses the stress distribution in a metal beam, which is modelled by the polynomial $P(x) = x^4 - x^3 - 7x^2 + x + 6$. Find all the roots of the…
A team of engineers analyses the stress distribution in a metal beam, which is modelled by the polynomial $P(x) = x^4 - x^3 - 7x^2 + x + 6$. Find all the roots of the polynomial to determine the exact points where the stress is zero.
Complete factorisation and repeated roots
Factorising "completely" means repeating common factor, identities and Ruffini's rule until all the factors are of degree $1$ (or of degree $2$ with no real roots). If a root appears more than once, it is a repeated root.
Given the already factorised polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly all of its roots and specify the multiplicity of each root (whether they are simple,…
Given the already factorised polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly all of its roots and specify the multiplicity of each root (whether they are simple, double, etc.).
Construct a polynomial of degree 3 that has roots $x_1 = -2$, $x_2 = 0$ and $x_3 = 4$, and whose leading coefficient is $1$. Express the final result in expanded form…
Construct a polynomial of degree 3 that has roots $x_1 = -2$, $x_2 = 0$ and $x_3 = 4$, and whose leading coefficient is $1$. Express the final result in expanded form (by multiplying out the factors).
Fully factorise the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$. To do this, first factorise out the common factor and then find the roots of the resulting polynomial.
Fully factorise the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$. To do this, first factorise out the common factor and then find the roots of the resulting polynomial.
Fully factorise the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$. Note that it may contain multiple roots. Clearly state its roots and the multiplicity of each one.
Fully factorise the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$. Note that it may contain multiple roots. Clearly state its roots and the multiplicity of each one.