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 $(x - 2)$. State clearly which is the quotient polynomial and which is the remainder of the division.

Without performing the division, use the remainder theorem to find the remainder when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.

Use the factor theorem to check if $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your work.

Factor out the common factor in the following polynomial, simplifying the expression fully:

$P(x) = 15x^5 - 10x^4 + 20x^3$

Factor the following polynomials by recognizing them as the expansion of algebraic identities (perfect square of a sum, perfect 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 factor it and find the other two roots.

Given the already factored polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly what all its roots are and specify the multiplicity of each one (whether they are simple, double, etc.).

Construct a polynomial of degree 3 that has $x_1 = -2$, $x_2 = 0$, and $x_3 = 4$ as roots, and whose leading coefficient is $1$. Express the final result in expanded form (multiplying out the factors).

A water tank has the shape of a rectangular prism. Its volume in terms of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of the tank is exactly $x$, factor the polynomial to find the algebraic expressions that correspond 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 test drone, in feet, with respect to the time $t$ in seconds since it begins its descent, is modeled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factor the polynomial to determine at what exact times the drone is at ground level ($H(t) = 0$).

Factor the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$ completely. To do this, first factor 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, factor the polynomial $P(x)$ completely.

The monthly profits of a company, in thousands of dollars, are given by the expression $B(x) = x^3 - 14x^2 + 49x$, where $x$ is the number of months elapsed since its creation. Factor the expression to find out in which month or months the company's profit is exactly zero.

Perform the division of 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 subsequently adjusting the quotient obtained.

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

A team of engineers analyzes the stress distribution in a metal beam, which is modeled 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 has a rectangular shape and its area is given by the polynomial $A(x) = x^4 - 81$. Factor the polynomial completely 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 $x_1 = 1$, $x_2 = -2$, and $x_3 = 3$ as roots, knowing also that its numerical value at $x = 0$ is $12$ (that is, $P(0) = 12$). Express the final polynomial in expanded form.

Factor the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$ completely. Bear in mind that it may contain multiple roots. State clearly what its roots are and the multiplicity of each one.

Using polynomial factorization 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. Check using the remainder theorem or Ruffini's rule that $x = 2$ is one of its roots. Next, factor the polynomial completely to find its other two roots (one of them is a fraction).

In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Factor the polynomial completely to discover all the roots and their respective multiplicities, thus managing to decipher the code.