Exercises: Factorisation and Ruffini's rule

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.

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.

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):

1. a) $A(x) = x^2 - 14x + 49$
2. b) $B(x) = 9x^2 - 16$

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 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 (by multiplying out the factors).

A water tank is in the shape of a rectangular prism. Its volume as a function of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of the tank is exactly $x$, factorise the polynomial to find the algebraic expressions corresponding to the dimensions of its rectangular base.

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 $(x - 2)$.

The height of a roller coaster car, in meters, relative to time $t$ in seconds from the start of a drop, is modeled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factor the polynomial to determine the exact times at which the car is at ground level ($H(t) = 0$).

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.

Calculate the value of $m$ so that the polynomial $P(x) = x^3 - 2x^2 + mx + 6$ is exactly divisible by $(x - 3)$. Once the value of $m$ is found, fully factorise the polynomial $P(x)$.

The number of visits to a website (in thousands) during month $x$ since its launch is given by the expression $V(x) = x^3 - 14x^2 + 49x$. Factorise the expression to find in which month or months the number of visits was exactly zero.

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.

A packaging company designs a cardboard box whose volume is given by the polynomial $V(x) = x^3 - 12x^2 + 36x$. Factorise the polynomial fully to express the volume as the product of three linear dimensions (length, width and height).

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.

Determine the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 4x + 4$ is simultaneously divisible by $(x - 1)$ and by $(x + 2)$.

A solar panel is rectangular in shape and its area is given by the polynomial $A(x) = x^4 - 81$. Fully factorise the polynomial into irreducible polynomials with real coefficients to find the possible algebraic expressions for its base and height.

Construct a polynomial $P(x)$ of degree 3 that has roots $x_1 = 1$, $x_2 = -2$ and $x_3 = 3$, knowing also that its value at $x = 0$ is $12$ (that is, $P(0) = 12$). Express the final polynomial in expanded form.

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.

Using polynomial factorisation and algebraic identities, prove algebraically that the difference between the squares of two consecutive odd numbers, $(2n+1)^2 - (2n-1)^2$, is always a multiple of $8$ for any integer $n$.

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).

In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Fully factorise the polynomial to find all the roots and their respective multiplicities, thereby deciphering the code.