Exercises: Polynomial factorisation

Factor out the common factor and use algebraic identities to factorize the following polynomial as much as possible:

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

Factorize the following third-degree 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 its factorized form.

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

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

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

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

Completely factorize 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, completely factorize the polynomial in the set of real numbers.

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

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

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

Simplify the following algebraic fraction by first factoring the numerator and the denominator completely:

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

Factorize the polynomial $P(x) = (x^2 - 4)^2 - (x + 2)^2$ using standard identities and extracting common factors, without needing to expand the whole 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 feet and $h(x)$ is the height in feet. Factorize $h(x)$ to find the exact points where the drone touches the ground ($h(x) = 0$).

Determine a fourth-degree polynomial, $P(x)$, which is monic (leading coefficient 1), has $x = -2$ as a double root, $x = 3$ as a simple root, and lacks a constant term. Write it in its factorized form and in its expanded general 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, factorize $P(x)$ completely.

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

Show that the polynomial $P(x) = x^4 + 4$ has no real roots. Then, factorize it as the product of two second-degree 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 completely factoring 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 investment in R&D in millions of dollars. Factorize the polynomial to find the exact break-even points (including fractional roots).

A third-degree 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, it is known that its constant term is $4$. Find the factorized expression of $P(x)$.

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

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 $g(x) = 7x^2 - 8x - 12$. Factorize the difference polynomial $D(x) = f(x) - g(x)$ to determine the $x$ coordinates of the pillars where both structures meet.