Exercises: Division of polynomials

Divide the following polynomial by the given monomial, simplifying each of the resulting terms:

In any polynomial division, the relation $P(x) = d(x) \cdot c(x) + r(x)$ holds. If when dividing a polynomial by $d(x) = x + 3$ we have obtained a quotient of $c(x) = x - 1$ and a remainder of $r(x) = 5$, what is the dividend polynomial $P(x)$?

Apply Ruffini's rule to calculate the quotient and the remainder of the following division:

Use Ruffini's rule to divide the polynomial $P(x) = x^3 + 4x^2 - x - 4$ by the binomial $d(x) = x + 1$. Clearly state the quotient $C(x)$ and the remainder $R$.

Without carrying out long division or using Ruffini's rule, calculate the value of the remainder of the following operation using the Remainder Theorem:

Calculate the value of the letter $k$ so that the division is exact (that is, the remainder is $0$):

The area of a rectangular recording studio is given by the polynomial $A(x) = 3x^2 + 15x$. If the width of the studio measures $3x$ metres, what is the algebraic expression that represents the length of the studio?

Check whether the number $a = 3$ is a root of the polynomial $P(x) = x^3 - 3x^2 + x - 3$ by carrying out division using Ruffini's rule and observing the remainder obtained. Justify your answer based on the result of the operation.

Elena wants to calculate the quotient and remainder of the division $(2x^3 - 7x + 5) : (x - 3)$ using Ruffini's rule. In her notebook, she has set up the table by writing the numbers $2$, $-7$, $5$ in the top row and, on the left, as the value associated with the divisor, the number $-3$.

1. a) Identify and explain the two conceptual mistakes Elena made before starting to operate: one concerning the sign of the divisor and the other concerning the coefficients of the dividend.
2. b) Correct both mistakes by properly redoing Ruffini's table, and explicitly write the obtained quotient polynomial $C(x)$ and remainder $R$.

Given the polynomial division $(6x^4 - x^3 - 2x^2 + 8x - 1) : (2x^2 - x + 1)$:

1. a) Before carrying out the algorithm, estimate and reason:

1. What exact degree will the quotient polynomial have?

2. What is the maximum degree the remainder can have? Could the remainder be a second-degree polynomial like $x^2 - 3$? Justify it according to the definition of polynomial division.

3. What will be the leading monomial (the term with the highest degree) of the quotient?

1. b) Perform the traditional long division using the box method algorithm and check whether your estimates about the degrees and the leading monomial match the obtained result.

Use polynomial long division to find the quotient $C(x)$ and the remainder $R(x)$ of the following polynomial operation. Remember to order and complete the terms of the dividend before starting:

$(6x^4 + 7x^3 - 5x^2 + 1) : (3x^2 - x + 2)$

Apply the Ruffini rule to calculate the quotient and the remainder of the division of the polynomial $P(x) = x^4 - 3x^2 + 5x - 10$ by the binomial $(x + 3)$.

Remember that, to apply the rule, the polynomial must be complete and ordered (if any term is missing, its coefficient is written as $0$) and that the value placed on the left is the root of the divisor.

1. a) Write the coefficients of the dividend and the value of $a$ that you use in the Ruffini rule.

1. b) Carry out the Ruffini rule step by step.

1. c) Write the quotient polynomial $C(x)$ and clearly state the value of the remainder $R$.

Without carrying out the division, use the Remainder Theorem to find out whether the number $x = 2$ is a root of the polynomial $P(x) = 2x^3 - 5x^2 - x + 6$. Justify your answer by indicating whether the division of $P(x)$ by $(x - 2)$ has no remainder (is exact) or not.

Find the value of the parameter $k$ so that the division of the polynomial $P(x) = x^3 - kx^2 + 4x - 8$ by the binomial $(x - 2)$ has a remainder equal to $10$.

The area of a rectangular padel court is given by the polynomial $A(x) = 2x^2 + 11x + 12$. If the width of the court measures $(x + 4)$ metres, determine the algebraic expression that represents its length.

Reconstruct the dividend polynomial $D(x)$ knowing that when divided by $d(x) = x^2 - 1$, the quotient obtained is $C(x) = x + 5$ and the remainder is $R(x) = 3x - 2$. Use the fundamental property of division: $D(x) = d(x) \cdot C(x) + R(x)$.

A team of young programmers is designing a level for a video game where the volume of a supply crate is $V(x) = x^3 - 6x^2 + 12x - 8$. If the base of the crate has an area of $A(x) = x^2 - 4x + 4$, what is the expression for the height of the crate? What value does the height take if $x = 5$ units?

Check whether the binomial $(x + 1)$ is a factor of the polynomial $P(x) = x^4 - 1$. To do this, you can use long division, Ruffini's rule, or the Factor Theorem. Explain what it means for a binomial to be a "factor" of another polynomial in terms of the remainder of the division.

Consider the polynomial $P(x) = x^4 - 3x^3 + 2x - 15$, and suppose you want to study its division by $(x - 3)$.

1. a) If the sole purpose of an exercise were to find out whether the division is exact (that is, if the remainder is zero), compare two possible approaches: Ruffini's rule and the Remainder Theorem. Justify which of the two procedures is faster and more efficient to answer that question.
2. b) Apply the method you selected as the most efficient in the previous section, calculate the remainder, and deduce with justification whether $(x - 3)$ is a factor of $P(x)$ or not.
3. c) If you were now required to find the quotient polynomial $C(x)$, reason whether the Remainder Theorem method would still be useful to you and obtain that quotient using the appropriate method.

We want to determine a third-degree polynomial $P(x)$ whose leading coefficient is $1$. The following information about its divisions is known:

1. When dividing it by $(x - 2)$, the division is exact.

2. When dividing it by $(x + 1)$, the remainder is $0$.

3. Its constant term is $-8$.

1. a) Using the first two conditions, determine two roots of $P(x)$ and write its expression in factored form: $P(x) = (x - 2) \cdot (x + 1) \cdot (x - a)$, where $a$ is an unknown root.
2. b) From the constant term, set up a simple equation to calculate the exact value of the root $a$ and expand the polynomial to express it in the standard form $P(x) = x^3 + Bx^2 + Cx + D$.
3. c) Use the Remainder Theorem (without doing long division or Ruffini's rule) to check whether the binomial $(x + 4)$ is also a factor of $P(x)$.

Find the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 - ax^3 + bx^2 + x - 6$ is divisible simultaneously by $(x - 1)$ and by $(x + 2)$. Hint: use the Remainder Theorem to set up a system of equations.

A logistics company uses cardboard boxes whose volume in cubic centimetres is given by the polynomial $V(x) = 2x^3 + 9x^2 + 7x - 6$. If the base area of the box is $B(x) = 2x^2 + 5x - 3$, calculate the polynomial that represents the height $H(x)$ of the box.

When dividing a polynomial $P(x)$ by $Q(x) = x^2 - 4$, we obtain a quotient of $C(x) = x^2 + 2x - 3$ and a remainder of $R(x) = 5x + 1$. Reconstruct the original polynomial $P(x)$ and express it in standard form (ordered from highest to lowest degree).

Determine the value of $k$ so that the remainder of the division $(2x^4 - 5x^3 + kx^2 - 12) : (x + 2)$ is equal to the remainder of the division $(x^3 + 2x^2 - x + 10) : (x - 3)$.

Calculate the quotient and the remainder of the following division using Ruffini's Rule. Pay attention, as the divisor does not have a leading coefficient equal to 1:

Remember that to use Ruffini's rule, the divisor must be of the form $(x - a)$. How could you transform $2x - 1$ to give it that form?

The total profit of a video game store from the sale of a new release is expressed by the polynomial $B(x) = x^3 + 15x^2 + 54x$. If the profit for each video game sold is represented by the binomial $(x + 6)$, perform the division to find the polynomial that represents the total number of video games sold.

A rectangular billboard is to be designed with a total surface area defined by $S(x) = x^4 - 16$. If one of its sides measures $L(x) = x^2 + 4$, calculate the polynomial that represents the other side. Next, state whether the area is divisible by $(x - 2)$ and justify your answer without carrying out long division.

Consider the polynomial $P(x) = 3x^3 - kx^2 + 12x - 4$.

1. Calculate the value of $k$ so that $x = 2$ is a root of the polynomial.

2. Once $k$ is found, use Ruffini's Rule to divide $P(x)$ by $(x - 2)$ and check that the remainder is zero.

3. Is the quotient obtained a monic quadratic polynomial? (A polynomial is monic if its leading coefficient is 1).

A parcel and logistics company based in Zaragoza designs modular boxes to optimize transport in vans. The total cargo volume (in cubic decimeters, equivalent to liters) corresponding to a shipment is modeled by the polynomial:

where $x$ is a design dimension in decimeters with $x > 1$. The surface area of the van's floor where the boxes are stacked, measured in square decimeters, is given by the polynomial:

1. a) Model the height $H(x)$ reached by the stack by dividing the volume $V(x)$ by the base area $S(x)$. Perform the division and interpret the meaning of the remainder obtained: do the boxes fit occupying the entire base without leaving gaps or leftover volume that prevents closing the tailgate?
2. b) If for a specific delivery $x = 3\text{ dm}$ is set, calculate the numerical value of the base area, the height, and the total volume, and check that the division algorithm check holds: $V(3) = S(3) \cdot H(3) + R$.

Consider a polynomial $P(x)$ whose full expression is unknown, but we know two results of its divisions: when dividing $P(x)$ by $(x - 2)$, the remainder obtained is $7$, and when dividing $P(x)$ by $(x + 1)$, the remainder obtained is $-2$.

1. a) If we divide $P(x)$ by the second-degree polynomial $d(x) = (x - 2) \cdot (x + 1) = x^2 - x - 2$, reason in detail from the relationship between degrees why the remainder $R(x)$ must have at most degree $1$ and can always be written as $R(x) = mx + n$.
2. b) State the fundamental division identity: $P(x) = (x^2 - x - 2) \cdot C(x) + (mx + n)$. Without knowing the quotient $C(x)$, substitute the appropriate values of $x$ into this identity to formulate a system of two linear equations in the unknowns $m$ and $n$.
3. c) Solve the system and write the final algebraic expression of the remainder $R(x)$.

Unit Challenge. How can shipping packaging be optimized? You are going to redesign a cardboard box for sustainable shipping starting from an everyday object.

1. Data collection and expressions: Measure in centimetres the length, width, and height of a real box you have at home. Consider that you increase the length by $x\text{ cm}$, the width by $(x - 1)\text{ cm}$, and the height remains fixed at its measured value $h$. Write the algebraic expressions for the new edges.

2. Product and sum: Determine the polynomial for the total volume $V(x)$ by multiplying the three dimensions and the polynomial for the total surface area $A(x)$ by adding the areas of its six faces to estimate the required cardboard.

3. Division: If the box will transport modular parts whose unit volume is $(x + 2)\text{ cm}^3$, divide $V(x)$ by $(x + 2)$ using polynomial division to obtain the quotient $C(x)$ and the remainder $R$, so that $V(x)=(x+2)C(x)+R$. For $x = 2,5$, check that the maximum number of complete units that fit by volume is the integer part of $V/(x+2)$ and explain why, in general, it does not coincide with the value of the quotient, nor the remainder with the unused volume.

4. Final report: Use a spreadsheet or GeoGebra to evaluate the usable volume and the cardboard used if it is enlarged by $x = 2,5\text{ cm}$, writing a reasoned proposal on the viability of the packaging.

Assessment criteria:

• Precision in formulating the algebraic expressions and calculating the product and sum.

• Mastery of the polynomial division algorithm and critical interpretation of quotient and remainder.

• Appropriate use of the digital tool and justification of conclusions based on the measured data.