Exercises: Roots of polynomials

Evaluate the polynomial $P(x) = x^3 - 4x^2 + x + 6$ to determine analytically whether the values $x = 2$ and $x = -1$ are roots of it.

Use Ruffini's rule to check whether $x = 3$ is a root of the polynomial $Q(x) = 2x^4 - 5x^3 - 4x^2 + 7x - 6$.

Write down all the integers that are candidates for being roots of the monic polynomial $R(x) = x^4 + 3x^3 - 5x^2 + x - 12$, based on its constant term.

Determine the roots of the already factorised polynomial $S(x) = 3x(x - 5)(x + 2)^2$, indicating which are simple roots and which are multiple roots.

Calculate the value of the parameter $k$ so that $x = -2$ is an exact root of the polynomial $T(x) = x^3 + kx^2 - 3x + 10$.

Construct a cubic polynomial, expanded and in descending powers, that has $x = 1$, $x = -3$ and $x = 4$ as exact roots, and whose leading coefficient is $1$.

Observe the graph of the polynomial function $y = P(x)$ and write down its real roots based on the points of intersection with the axes.

Justify, without performing any complex calculation or using Ruffini's rule, why $x = 0$ must necessarily be a root of the polynomial $V(x) = 5x^5 - 7x^3 + 2x^2 - 9x$.

In Mathematics class, Lucía is looking for the integer roots of the polynomial $P(x) = x^3 - 3x^2 - 4x + 12$ and writes the following conclusion on the board: "Since the constant term is $12$, the only possible integer roots are its positive divisors: $1$, $2$, $3$, $4$, $6$, and $12$. I have checked that substituting $x = 2$ gives $P(2) = 0$, so I conclude that the binomial $(x + 2)$ is a factor that divides $P(x)$ exactly". Identify the two conceptual errors that Lucía has made, explain in detail why they contradict the relationship between roots, divisors, and factors, and write the two corrected statements.

At an AEMET weather station in Teruel, the temperature $T$ (in degrees Celsius) during a winter morning is modeled as a function of the elapsed time $t$ (in hours elapsed since 8:00 h, in the interval $t \in [0, 6]$) by the polynomial $T(t) = -t^2 + 5t - 6$.

1. a) Find the roots of $T(t)$ analytically and explain their physical meaning on the station's thermometers.
2. b) At what exact clock times was the temperature $0\text{ }^\circ\text{C}$?
3. c) Knowing that between both roots the graph does not cross the horizontal axis again, evaluate the polynomial at an intermediate time value to determine with reasoning whether during that time interval the temperature was above freezing or below zero.

Find all integer roots of the quartic polynomial $P(x) = x^4 - 2x^3 - 7x^2 + 8x + 12$ by using Ruffini's rule iteratively.

The profit of a technology company, in thousands of euros, is given by the polynomial $B(x) = -x^3 + 6x^2 - 5x$, where $x$ represents the months elapsed since the beginning of the financial year. Determine in which exact months the company's profit is zero.

Find the values of the parameters $a$ and $b$ so that the polynomial $Q(x) = x^3 + ax^2 + bx - 6$ has $x = 1$ and $x = -2$ as exact roots.

An engineer designs a metal part whose volume, in cubic centimetres, depends on an adjustment measure $x$ according to the expression $V(x) = 2x^3 - 9x^2 + 10x - 3$. Determine which values of $x$ make the volume zero (that is, find all roots of the polynomial).

Write a quartic polynomial, in expanded form, that has $x = 2$ as a double root, $x = -1$ as a simple root, has no constant term, and whose leading coefficient is $3$.

Determine all rational roots of the polynomial $R(x) = 3x^3 + 5x^2 - 4x - 4$. To do so, first state the complete list of candidate fractions and then check which ones make the polynomial zero.

The vertical trajectory of an underwater drone during testing follows the curve given by $A(t) = t^4 - 10t^3 + 35t^2 - 50t + 24$, where $A$ is the depth in metres and $t$ is the time in minutes from the start of the dive. Find at which times $t$ the drone is exactly at the surface of the water ($A(t) = 0$).

Prove with reasoning that the polynomial $P(x) = x^4 + 3x^2 + 5$ has no real roots by analysing the signs of its terms for any value of $x \in \mathbb{R}$.

An olive oil cooperative in Jaén analyzes its daily profit margin $B(x)$ (in hundreds of euros) when processing $x$ tonnes of olives using the function $B(x) = -x^3 + 8x^2 - 11x - 20$, with $x \ge 0$. The break-even threshold (break-even point where neither money is gained nor lost) is reached precisely at the roots of $B(x)$.

1. a) Before calculating anything, explain why the negative divisors of the constant term lack economic meaning in this context. If the technical capacity of the olive oil mill does not exceed $6$ tonnes per day, estimate which possible positive integer root candidates make sense to evaluate.
2. b) Justify and choose the candidate you consider most promising, and verify by substitution or by Ruffini's rule whether it is a root of $B(x)$.
3. c) Factor the polynomial to find all roots with economic meaning and interpret with what daily processing volume the mill operates at the break-even point without losses.

A video game programmer in Valencia is designing the terrain for a two-dimensional platformer level, where ground level is given by the horizontal $X$-axis. The design requires a monic polynomial $P(x)$ (with leading coefficient $1$) that satisfies the following conditions:

1. The character crosses ground level by passing through it at $x = -1$ and at $x = 3$.

2. At $x = 2$, the terrain touches ground level, bouncing upward without sinking (that is, $x = 2$ is a root of multiplicity $2$).

3. The polynomial has the lowest possible degree that allows these characteristics.

1. a) Write the factored form of $P(x)$ and deduce its degree.
2. b) Without needing to multiply all the factors, calculate the constant term of $P(x)$ with reasoning, explaining how it relates to the given roots.
3. c) If the designer decides to multiply the entire polynomial by $3$ to triple the height of the hills, would the points where the terrain touches the ground change? Justify your answer using the definition of a root.

Knowing that a monic cubic polynomial $P(x) = x^3 + ax^2 + bx + c$ has roots $x_1 = -1$, $x_2 = 2$ and $x_3 = 4$, determine the numerical value of its constant term $c$ and its degree-two coefficient $a$. Justify your answer by deducing it from the factorised form without performing the full multiplication of the three binomials.

An open-topped box is made from a rectangular sheet of cardboard measuring $40\text{ cm}$ by $30\text{ cm}$, by cutting identical squares of side $x$ from the four corners and folding up the flaps. Find the polynomial $V(x)$ defining the volume of the box and determine all its mathematical roots (including those that lack physical meaning for the construction of the box).

Find the expression for a degree 5 polynomial $Q(x)$ that simultaneously satisfies all of the following conditions:

• $x = \sqrt{2}$ and $x = -\sqrt{2}$ are simple roots.

• $x = 3$ is a double root.

• The graph of the polynomial passes through the origin.

• The numerical value of the polynomial at $x = 1$ is $Q(1) = 8$.

It is known that the polynomial $R(x) = x^4 - 4x^3 + x^2 + 6x + k$ has a root at $x = 2$. First, find the value of the parameter $k$. Next, check using Ruffini's rule whether $x = 2$ is a multiple root (double, triple...) and find the remaining real roots of the polynomial.

In a study of population dynamics, the growth of a bacterial colony is given by the model $C(t) = t^4 - 8t^3 + 22t^2 - 24t + 9$. A biologist claims that the population becomes zero at exactly two distinct moments in time, both with even multiplicity. Factorise the polynomial to check whether the biologist is right and state the times at which $C(t) = 0$.

Prove analytically that if a polynomial $P(x) = a_n x^n + \dots + a_1 x + a_0$ has all integer coefficients and both $P(0)$ and $P(1)$ result in odd numbers, then $P(x)$ cannot have any integer roots. (Hint: reason about the parity of $x$ when it is even and when it is odd).

The production cost of a technological component follows the polynomial function $C(x) = 8x^3 - 36x^2 + 46x - 15$. Knowing that $x = \frac{3}{2}$ is one of the exact roots of this polynomial, determine the other two roots by solving the equation resulting after the division.

A 3D designer creates a curved surface whose cross-sectional profile is modelled by the polynomial $P(x) = x^5 - 2x^4 - 6x^3 + 8x^2 + 5x - 6$. Find all roots of this profile to determine the exact points where it intersects the horizontal reference axis.

To find out which of the numbers $x = -1$, $x = 2$, and $x = 5$ are roots of the polynomial $P(x) = x^4 - 2x^3 - 13x^2 + 14x + 24$, two students propose different approaches:

• Sergio suggests applying Ruffini's rule directly and successively with each of the three numbers.

• Marta proposes looking first at the divisors of the constant term and, for the values that are not ruled out, calculating the numerical value $P(a)$.

1. a) Critically compare both methods: what advantages and disadvantages does each have depending on whether one only seeks to test the roots or also to factor the polynomial?
2. b) Applying the integer root theorem to the constant term, justify why one of the three proposed values can be ruled out immediately without performing any long arithmetic operations.
3. c) Select and apply the most convenient strategy to verify which of the remaining values are indeed roots of $P(x)$, and obtain the resulting quotient polynomial after extracting those factors to find the missing roots.

The Dirección General de Tráfico (DGT) and IDAE are studying the fuel consumption of delivery vans on interurban roads. For a typical vehicle, the deviation in fuel consumption relative to the optimal reference level (in liters per $100\text{ km}$) is described by the polynomial $C(v) = v^3 - 27v^2 + 236v - 660$, where $v$ represents the driving speed in tens of kilometers per hour (that is, $v = 6$ corresponds to $60\text{ km/h}$, $v = 10$ to $100\text{ km/h}$, etc.).

1. a) Knowing that at $100\text{ km/h}$ ($v = 10$) the vehicle reaches exactly one of its reference consumption levels ($C(10) = 0$), verify using Ruffini's rule that $v = 10$ is a root and factor $C(v)$ into a product of three first-degree factors.
2. b) Find all the real roots of the polynomial and interpret which of them apply to real driving according to the legal speed limits on Spanish dual carriageways and motorways ($60\text{ km/h}$ minimum speed and $120\text{ km/h}$ maximum speed).
3. c) Without calculating the expanded numerical value of $C(8)$, use the individual sign of each of the three factors obtained in part a) to reason whether driving at $80\text{ km/h}$ ($v = 8$) puts the vehicle's consumption above or below the optimal reference level.