Exercises: Factorisation of polynomials

Factorise out the common factor and use algebraic identities to fully factorise the following polynomial:

$P(x) = 3x^5 - 12x^3$

Factorise the following cubic polynomial knowing that one of its roots is $x = 2$:

$P(x) = x^3 - 7x^2 + 16x - 12$

Find the integer roots of the polynomial $P(x) = x^3 + 2x^2 - 5x - 6$ using Ruffini's rule and express it in factorised form.

Write the polynomial $P(x) = x^4 - 81$ in factorised form by applying the difference of two squares identity successively.

State the roots of the polynomial $P(x) = 5(x+3)^2(x-1)(x^2+4)$ and their respective multiplicities. Justify why this polynomial has no other real roots.

Factorise the polynomial $P(x) = 2x^2 + 5x - 3$ by first solving the associated quadratic equation.

Check, using the Remainder Theorem, whether $x = -1$ is a root of the polynomial $P(x) = x^4 + 3x^3 - 2x^2 + x + 7$. If so, factorise out the corresponding factor.

Fully factorise the biquadratic polynomial $P(x) = x^4 - 5x^2 + 4$.

Seán needs to factor the polynomial $P(x) = 3x^3 - 12x^2 + 12x$. On his exam, he writes the following procedure:

1) Factors out the common factor: $P(x) = x(3x^2 - 12x + 12)$.

2) Solves the equation $3x^2 - 12x + 12 = 0$ using the quadratic formula and obtains a single double root: $x = 2$.

3) Concludes that the final factorization is: $P(x) = x(x - 2)^2$.

1. a) Without expanding the entire polynomial, point out which immediate property fails in Seán's result (hint: compare the leading coefficient of his answer with that of the original polynomial).
2. b) Explain with reasoning the conceptual error he made when converting the solutions of the quadratic equation into factors of the polynomial.
3. c) Correct the procedure and write the complete and correct factorization of $P(x)$.

To factor the polynomial $P(x) = 2x^4 - 32$ into irreducible factors over $\mathbb{R}$, two students suggest different strategies:

• Aoife proposes applying Ruffini's rule successively by testing integer divisors of $32$ until the degree is reduced.

• Niamh proposes factoring out a numerical common factor first and then applying algebraic identities (differences of squares) repeatedly.

1. a) Compare both methods and justify which of the two is more efficient, faster, and less prone to calculation slips in this case.
2. b) Apply the chosen strategy step by step until you obtain the complete factorization of $P(x)$ over $\mathbb{R}$.
3. c) Justify with reasoning why the quadratic factor $(x^2 + 4)$ cannot be factored further over the real numbers.

Determine the value of the parameter $m$ so that the polynomial $P(x) = x^3 - mx^2 + 4x - 4$ is divisible by the binomial $x - 2$. Once the value of $m$ is found, fully factorise the polynomial over the set of real numbers.

The volume of a rectangular box as a function of its length $x$ is given by the polynomial expression $V(x) = x^3 - 7x^2 + 10x$. Factorise $V(x)$ to find the simplest algebraic expressions representing the three dimensions of the box.

Factorise the polynomial $P(x) = 3x^4 + 5x^3 - 17x^2 - 13x + 6$. Note that some of its roots may be fractional numbers.

A biologist models the growth and decline of a bacterial population using the polynomial $N(t) = -t^4 + 10t^3 - 35t^2 + 50t - 24$, where $t$ is the time in hours. Factorise the polynomial to determine the exact times at which the population becomes zero.

Simplify the following algebraic fraction by fully factorising the numerator and the denominator first:

$\frac{x^3 - 3x^2 - 4x + 12}{x^3 - 5x^2 + 6x}$

Factorise the polynomial $P(x) = (x^2 - 4)^2 - (x + 2)^2$ using algebraic identities and factorising out common factors, without needing to expand the entire polynomial first.

The trajectory of a racing drone in an exhibition section follows the polynomial profile $h(x) = 2x^4 - 14x^3 + 30x^2 - 18x$, where $x$ is the horizontal distance in metres and $h(x)$ is the height in metres. Factorise $h(x)$ to find the exact points where the drone touches the ground ($h(x) = 0$).

Determine a fourth-degree polynomial, $P(x)$, that is monic (leading coefficient 1), has $x = -2$ as a double root, $x = 3$ as a simple root, and has no constant term. Write it in its factorised form and in its general expanded form.

Consider the polynomial with integer coefficients $P(x) = 2x^3 + 3x^2 - 8x + 3$.

1. a) Before applying any division algorithm, make a reasoned preliminary estimate:

• List all possible rational root candidates $\frac{p}{q}$ according to the rational root theorem.

• Calculate mentally or with quick trial the numerical value at the simplest integer values ($x = 1$, $x = -1$, $x = 0$) and estimate how many non-integer rational roots you expect to find at most.

1. b) Factor $P(x)$ completely over $\mathbb{R}$ using Ruffini's rule and the associated quadratic equation.
2. c) Check whether the roots obtained match your list of rational candidates and verify whether the product of the three roots obtained $r_1 \times r_2 \times r_3$ is consistent with the value of the quotient $-\frac{a_0}{a_n} = -1.5$.

Find a third-degree polynomial $P(x)$ with integer coefficients that simultaneously satisfies the following three conditions:

1) It has a double root at $x = -2$.

2) Its constant term is $-12$ (that is, the graph of the associated polynomial function intersects the y-axis at the point $(0, -12)$).

3) When dividing $P(x)$ by the binomial $(x - 1)$, the remainder obtained is $-9$.

1. a) Set up the structure of the polynomial in terms of its roots and find its coefficients with reasoning.
2. b) Write the factored expression of $P(x)$, indicating each of its roots and their respective multiplicities.
3. c) Expand the obtained polynomial to verify that all its coefficients are integers and that it satisfies the remainder condition when divided by $(x - 1)$ using the remainder theorem.

Calculate the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 12x + 8$ is divisible by the polynomial $Q(x) = x^2 - 4$. Then, fully factorise $P(x)$.

The design profile of a precision mechanical part is defined by the polynomial function $P(x) = x^5 - x^4 - 7x^3 + x^2 + 6x$. Factorise the polynomial to find the exact coordinates of the anchoring points of the part on the horizontal axis.

Prove that the polynomial $P(x) = x^4 + 4$ has no real roots. Then, factorise it as the product of two quadratic polynomials. Hint: Add and subtract $4x^2$ to the expression to force the appearance of a difference of two squares.

Simplify the following algebraic expression, which represents the equivalent impedance in a complex circuit, by fully factorising each polynomial in the numerators and denominators:

$\frac{2x^3 - 7x^2 + 7x - 2}{2x^3 - x^2 - 8x + 4} - \frac{x^2 - 1}{x^2 + x - 2}$

In an economics study, the marginal profit curve of a technology company fits the polynomial $B(x) = 6x^4 - 11x^3 - 26x^2 + 21x + 10$, where $x$ represents the R&D investment in millions of euros. Factorise the polynomial to find the exact equilibrium points (including fractional roots).

A cubic polynomial $P(x)$ satisfies $P(1) = 0$, $P(-2) = 0$, and when divided by the binomial $(x-2)$ the remainder of the division is $16$. Furthermore, its constant term is known to be $4$. Find the factorised expression for $P(x)$.

Fully factorise the polynomial $P(x) = x^6 - 2x^5 - 3x^4 + 8x^3 - 4x^2$. Clearly identify its real roots and the multiplicity of each of them.

The outline of the vault of an avant-garde building is given by the geometric intersection of the curves $f(x) = x^4 - 2x^3$ and $g(x) = 7x^2 - 8x - 12$. Factorise the difference polynomial $D(x) = f(x) - g(x)$ to determine the $x$ coordinates of the pillars where both structures join.

An agricultural cooperative in Kilkenny orders the manufacture of a rectangular cuboid steel tank to store cold-pressed rapeseed oil. Due to space constraints in the oil mill, the dimensions of the tank (in meters) must satisfy the following relationships:

• The width is $x$ meters.

• The length must exceed the width by $2\text{ m}$.

• The height must be equal to the sum of the length and width, decreased by $1\text{ m}$.

• The total capacity required is exactly $40\text{ m}^3$ ($40,000\text{ liters}$).

1. a) Derive the polynomial expression for the volume $V(x)$ and set up the equation $V(x) - 40 = 0$ that allows you to determine the width $x$.
2. b) Before factoring, define the physical domain of the problem: why do we discard in advance any root less than or equal to zero?
3. c) Factor the polynomial $V(x) - 40$ completely over $\mathbb{R}$, explaining which rational candidates you test using Ruffini's rule.
4. d) Determine the real dimensions of the tank (width, length, and height) and interpret the role of the remaining quadratic factor: why does its discriminant guarantee that the solution found is geometrically and physically unique?

We want to factor the degree six polynomial $P(x) = x^6 - 9x^3 + 8$ into irreducible factors over $\mathbb{R}$.

In the classroom, two solution approaches are suggested:

• Method A (Seán): Treat it as a general degree 6 polynomial and apply Ruffini's rule successively, searching for integer roots among the divisors of $8$.

• Method B (Aoife): Treat it as a polynomial reducible to a quadratic by using the change of variable $t = x^3$, factor the quadratic equation in $t$, and finally factor each resulting cubic factor using algebraic identities (specifically, the difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$).

1. a) Analyze Seán's proposal: show at what exact point this method would get stuck if Seán only looks for integer roots using Ruffini, and why this algorithm alone is unable to factor quadratic factors that are irreducible over $\mathbb{R}$.
2. b) Work out Aoife's Method B completely until you obtain the factorization of $P(x)$ as a product of degree 1 and degree 2 factors.
3. c) Formally prove, by analyzing the discriminant of each obtained second-degree factor, that these factors are irreducible over $\mathbb{R}$ and that, therefore, the factorization cannot proceed any further.