Exercises: Division of polynomials

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

In any polynomial division, the relation $P(x) = d(x) \times c(x) + r(x)$ 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 $3x$ meters, 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.

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 $(x + 4)$ meters, 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: $D(x) = d(x) \times C(x) + R(x)$.

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.

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 centimeters 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).