Exercises: Polynomials. Factorisation

In designing a new level for a video game, the energy of a protective shield follows the polynomial function $P(x) = x^3 - 5x^2 + 7x - 3$, where $x$ represents the time in seconds since activation. Use the Remainder Theorem to check whether the shield runs out of energy at exactly $x = 3$ seconds.

Simplify the following expression by taking out the highest common factor from the terms of the polynomial that represents the revenue of a small custom t-shirt business:

$I(x) = 12x^5 - 18x^4 + 6x^2$

Expand the following expressions using algebraic identities. These expressions often appear when calculating variations in area on architectural plans:

1. a) $(2x + 5)^2$
2. b) $(3x - 2)^2$
3. c) $(x + 7)(x - 7)$

A farmer has a square plot and has decided to enlarge it. The area of the new plot, which is also square, is given by the trinomial $A(x) = x^2 + 10x + 25$, where $x$ is a length in meters and the area is in square meters. Factor this expression using special identities (square of a sum) to find the length of the side of the new plot in terms of $x$.

Apply Ruffini's rule to factorise the polynomial $P(x) = x^3 - 6x^2 + 11x - 6$, knowing that $x = 1$ is one of its roots. Write the polynomial as a product of three linear factors.

Determine the value of the unknown coefficient $k$ so that the polynomial $P(x) = x^3 - 2x^2 + kx - 10$ is divisible by $(x - 5)$. Remember that to be divisible, the remainder of the division must be zero.

A rectangular display screen is being designed whose surface area in square metres is described by the polynomial $S(x) = x^2 - 16$.

1. a) Factorise the polynomial using the difference of two squares identity.
2. b) If $x$ must be greater than 4 for the dimensions to make sense, what would be the expressions for the base and height of the screen?

Write the expression for a monic polynomial (one whose leading coefficient is 1) of degree three knowing that its roots are $x = 0$, $x = 2$ and $x = -3$. Express the final result as a product of factors.

Aoife wants to find out if the binomial $(x + 2)$ is a factor of the polynomial $P(x) = x^3 - 3x^2 - 4x + 12$. To check it without having to perform long division, she decides to use the Remainder Theorem and calculates:

$P(2) = 2^3 - 3\times 2^2 - 4\times 2 + 12 = 8 - 12 - 8 + 12 = 0$

Observing that the obtained value is zero, she concludes: "Since $P(2) = 0$, it is proven that $(x + 2)$ divides $P(x)$ exactly and, therefore, is one of its factors."

1. a) Identify the conceptual mistake Aoife makes and explain why her calculation does not prove that $(x + 2)$ is a factor of $P(x)$.
2. b) Correctly apply the Remainder Theorem to determine whether $(x + 2)$ is or is not a factor of said polynomial.
3. c) In light of the calculations performed, reasonably indicate two first-degree binomials of the form $(x - a)$ that are indeed factors of $P(x)$ with certainty.

To factor the polynomial $P(x) = 3x^3 - 12x$ into irreducible factors over $\mathbb{R}$, two classmates propose different strategies:

• Cian proposes to directly apply Ruffini's rule by completing the polynomial as $3x^3 + 0x^2 - 12x + 0$ and testing with the integer divisors of the constant term.

• Aoife argues that Ruffini should not be used in this case: she defends that it is much faster and safer to first factor out the common factor and then apply a notable identity (difference of squares).

1. a) Reasonably compare both proposals: point out the disadvantages of Cian's method compared to Aoife's proposal (in terms of computational effort and risk of error) and justify which is the optimal procedure.
2. b) Apply the most efficient strategy to obtain the complete factorization of $P(x)$ and indicate all its real roots.

Completely factorise the following polynomial by decomposing it into the product of as many linear factors as possible:

$P(x) = 2x^4 - 4x^3 - 10x^2 + 12x$

To do so, factorise by taking out the common factor, use Ruffini's rule and, if necessary, solve the resulting quadratic equation.

Calculate the value of the parameter $k$ so that the polynomial $P(x) = x^3 - kx^2 + 5x + 12$ is divisible by the binomial $(x + 1)$. Once the value of $k$ is found, find the remaining roots of the polynomial.

Determine the algebraic expression of a cubic polynomial $P(x)$ knowing that it satisfies the following conditions:

• Its roots are $x = -2$, $x = 1$ and $x = 3$.

• The value of the polynomial for $x = 0$ is $12$ ($P(0) = 12$).

Express the final result in expanded form (as a sum of monomials).

An architect designs a scale model of a logistics warehouse in the shape of a rectangular prism. The volume of the model is given by the polynomial $V(x) = x^3 + 7x^2 + 6x$, where $x$ represents the height in decimetres.

1. a) Factorise the polynomial $V(x)$ to obtain the expressions for the other two dimensions (length and width) in terms of $x$.
2. b) If the height of the model is $2$ dm, what are the dimensions of the base and what is the total volume?

Find the highest common factor (HCF) and the lowest common multiple (LCM) of the following polynomials:

$P(x) = x^3 - 4x$

$Q(x) = x^3 - 4x^2 + 4x$

A financial services company analyses the trend of its monthly profits using a polynomial function. It is known that the profits $B(t)$ (in thousands of euros) during the first $6$ months of the year follow the model:

$B(t) = t^3 - 9t^2 + 23t - 15$

Where $t$ is the time in months ($t=1$ is January, $t=2$ is February, etc.). Factorise the polynomial $B(t)$ and determine in which months the company's profit was exactly zero.

Factorise the following polynomials using algebraic identities and taking out common factors:

1. a) $R(x) = 4x^3 - 12x^2 + 9x$
2. b) $S(x) = x^4 - 81$
3. c) $T(x) = 3x^2 + 30x + 75$

Apply the Rational Root Theorem to find all real roots of the polynomial $P(x) = 2x^3 + 3x^2 - 8x + 3$. Once found, write its factorisation.

Consider the polynomial $P(x) = x^4 - 15x^2 - 10x + 24$.

1. a) Estimate before calculating:

• If $P(x)$ is divided by $(x - 10)$, will the remainder be positive or negative? Without calculating the exact numerical value, estimate its order of magnitude (tens, hundreds, thousands...) by reasoning which of the terms of the polynomial clearly dominates over the others when $x = 10$.

• Using the Rational Root Theorem, argue why values such as $x = 5$ or $x = 7$ are automatically ruled out as possible integer roots of this polynomial without having to substitute them or use Ruffini's rule.

1. b) Calculate and check:

• Determine the exact value of the remainder of the division $P(x) : (x - 10)$ by applying the Remainder Theorem and check if it agrees with your previous estimate.

• Find all the integer roots of $P(x)$ by completely factoring it over $\mathbb{R}$ using Ruffini's rule and subsequently solving the resulting quadratic equation.

The financial department of a company needs to define a degree 4 polynomial $P(x)$ that simultaneously meets four specifications to model a profitability index:

1. It is divisible by the irreducible quadratic polynomial $(x^2 - 2x + 5)$.

2. It vanishes at the values $x = -1$ and $x = 3$.

3. When divided by $(x - 1)$, the remainder obtained is exactly $-32$.

4. Its constant term in expanded form must be a negative integer.

1. a) Deduce the factored expression of $P(x)$, calculating with justification the leading coefficient necessary to satisfy the remainder condition.
2. b) Expand the obtained polynomial in decreasing powers of $x$, expressing it in the form $P(x) = ax^4 + bx^3 + cx^2 + dx + e$.
3. c) Reason whether there exists any other degree 4 polynomial that satisfies the first three conditions but whose constant term is positive.

A team of cybersecurity experts uses polynomials to generate encryption keys. The security of a bank account depends on the coefficients of the polynomial $P(x) = x^4 + ax^3 + bx^2 - 29x + 30$. For the key to be valid, the polynomial must be divisible by the binomial $(x - 2)$ and, additionally, the remainder of dividing $P(x)$ by $(x + 1)$ must be exactly $48$.

Calculate the values of parameters $a$ and $b$ for the key to be secure and, subsequently, find the complete factorisation of the resulting polynomial over the set of real numbers.

A logistics company is designing a new model of folding cardboard box to reduce plastic use. The volume of the box, in cubic decimetres, is given by the polynomial $V(x) = x^3 - 13x^2 + 50x - 56$, where $x$ represents one of the main dimensions in decimetres.

Factorise the polynomial to find the expressions for the three dimensions of the box (length, width and height) in terms of $x$. Knowing that all dimensions must be greater than $1$ dm, determine the range of possible values for the variable $x$.

In a study on population biology, the growth of a bacterial colony has been modelled using the degree six polynomial $B(t) = t^6 - 9t^3 + 8$, where $t$ is the time in hours. Biologists need to know at which exact moments the population is zero to understand the extinction cycles.

Factorise the polynomial $B(t)$ using a suitable change of variable to convert it into a quadratic polynomial. Express the result as a product of linear factors and irreducible quadratic factors over the set of real numbers.

A surveillance drone follows a trajectory whose height above sea level is described by the polynomial $H(x) = x^3 - 12x^2 + kx - 48$. It has been observed that the drone touches the ground (height $H(x) = 0$) at three different points whose distances to the origin of coordinates, $x_1, x_2$ and $x_3$, form an arithmetic progression (that is, the difference between consecutive terms is constant).

Find the value of the constant $k$ and use the Factor Theorem to find the three distances at which the drone lands.

Consider an unknown polynomial $P(x)$ for which we only have partial information obtained through computer lab tests. We know that when dividing $P(x)$ by $(x - 3)$, the remainder obtained is $10$. On the other hand, when dividing the same polynomial $P(x)$ by $(x + 2)$, the remainder is $-5$.

Determine, giving reasons, what the remainder of dividing the polynomial $P(x)$ by the product $(x - 3)(x + 2)$ will be. Hint: Remember that the degree of the remainder is always less than the degree of the divisor.

The weekly net profit of a vintage clothing startup is modelled by the function $B(x) = -x^3 + 15x^2 - 44x + 30$, where $x$ is the number of thousands of euros invested in digital marketing.

To conduct an audit, the managers need to know the break-even points (where profit is zero). Knowing that for an investment of $1000$ euros ($x=1$) the profit is already zero, completely factorise the polynomial to find all investment values that produce zero profit. Interpret whether there are values of $x$ for which the company makes a loss based on the factorisation.

Factorise the polynomial $P(x) = 12x^3 + 8x^2 - 13x + 3$ knowing that it has no integer roots. You must apply the Rational Root Theorem to identify the possible candidates of the form $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Once the first rational root has been found, use Ruffini's rule and solve the resulting quadratic equation to find the other roots and write the complete factorisation.

It is desired to build a public park with a central fountain having a rectangular base. The total area allocated for the fountain is $A(x) = 2x^4 - 20x^2 + 18$ square metres. The architects need the area to be positive for the design to be feasible.

Factorise the polynomial $A(x)$ as a product of four linear factors. Use this factorisation to determine the intervals of values of $x$ (with $x > 0$) for which the area is positive and, therefore, the fountain can be built.

A cooperative oil mill in Kilkenny analyzes the launch of a line of extra virgin olive oil. The economic team estimates that the quarterly net profit $B(x)$ (expressed in thousands of euros) depends on the selling price $x$ (in euros per liter) according to the third-degree polynomial:

$B(x) = -x^3 + 14x^2 + 35x - 300$

where the permissible market price lies in the interval $[0, 20]$ euros per liter.

1. a) If the cooperative set a price of $0$ €/l ($x = 0$), what accounting result would it obtain? Interpret the economic meaning of the constant term within the company's cost structure (fixed costs).
2. b) Knowing that setting a price of $4$ €/l places the company exactly at the break-even point (zero profit), use Ruffini's rule and the resolution of the resulting equation to completely factor $B(x)$ over $\mathbb{R}$.
3. c) Determine all the selling prices that yield zero profit ($B(x) = 0$) and interpret which roots lack economic meaning in the context of the real market.
4. d) Study the sign of $B(x)$ in the intervals bounded by its roots within $[0, 20]$ and reasonably advise management: in what price range does the cooperative obtain strictly positive net profits ($B(x) > 0$)? Explain why the profit becomes negative again if the price exceeds $15$ €/l.

Cian must completely factor the polynomial $P(x) = 2x^3 - x^2 - 7x + 6$ over $\mathbb{R}$. In his work, he follows these steps:

1. He checks that $x = 1.5 = \dfrac{3}{2}$ is a root by calculating $P(1,5) = 0$.

2. He applies Ruffini's rule with the value $1.5$ on the coefficients $[2, -1, -7, 6]$ and obtains the coefficients $[2, 2, -4]$ in the bottom row with a final remainder of $0$.

3. He solves the equation $2x^2 + 2x - 4 = 0$, obtaining the roots $x = 1$ and $x = -2$, so he factors that quotient as $2(x - 1)(x + 2)$.

4. He writes the final factorization of $P(x)$ by combining the divisor associated with the first root with the factors found:

$P(x) = (2x - 3)\times 2(x - 1)(x + 2) = 2(2x - 3)(x - 1)(x + 2)$

However, when multiplying the factors to check his work, he is puzzled to observe that the highest-degree term turns out to be $4x^3$ and not $2x^3$.

1. a) Locate with mathematical accuracy the error made by Cian in step 4 using the division algorithm identity $D(x) = d(x)\times C(x) + R$.
2. b) Explain what quotient Ruffini's rule actually provides when a fractional number of the form $\dfrac{a}{b}$ is used, and what algebraic adjustment is essential if one wants to express the division with respect to the binomial $(bx - a)$.
3. c) Write the correct and complete factorization of $P(x)$ as a product of irreducible linear factors over $\mathbb{R}$ and verify that the leading coefficient matches the original one.