Polynomials. Factoring

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 $x$.

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) $T(x) = 3x^2 + 30x + 75$

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:

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

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

1. a) Factorize 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?

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 decimeters.

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?

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.

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.

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

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 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.

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.

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.

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.

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).

A financial services company analyzes the trend of its monthly profits using a polynomial function. It is known that the profits $B(t)$ (in thousands of pesos) 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.). Factor the polynomial $B(t)$ and determine in which months the company's profit was exactly zero.

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.

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.

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.

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 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.

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.

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 meters. The architects need the area to be positive for the design to be feasible.

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

The weekly net profit of a vintage clothing startup is modeled by the function $B(x) = -x^3 + 15x^2 - 44x + 30$, where $x$ is the number of thousands of pesos 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 $1000$ pesos ($x=1$) the profit is already zero, completely factorize 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 factorization.

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

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

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