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 $P(x) = x^4 - 5x^2 + 4$.

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 meters and $h(x)$ is the height in meters. Factor $h(x)$ 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.

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 pesos. Factorize 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.