MTH120 College Algebra Chapter 5.4

 5.4 Dividing Polynomials:

Dividing polynomials is a fundamental operation in algebra, similar to dividing numbers. To divide one polynomial by another, you can use long division or synthetic division, depending on the situation. Here, we'll cover the process of dividing polynomials using long division:

Long Division of Polynomials:

Long division of polynomials is similar to long division of numbers. You divide the polynomial (the dividend) by another polynomial (the divisor) and obtain a quotient and a remainder. Here's how to do it:

Step 1: Arrange the Polynomials

Write the dividend (the polynomial you're dividing) inside the long division symbol, and write the divisor (the polynomial you're dividing by) on the outside.

        divisor
____________________
dividend

Step 2: Divide the First Term

Divide the first term of the dividend by the first term of the divisor and write the result as the first term of the quotient. This is similar to how you divide the first digit of a number.

        divisor
____________________
quotient term1
dividend

Step 3: Multiply and Subtract

• Multiply the entire divisor by the term you just found in the quotient.
• Write the result below the dividend.
• Subtract this result from the dividend, and write the difference underneath.

        divisor
____________________
quotient term1
dividend - (divisor * quotient term1)

Step 4: Bring Down the Next Term

Bring down the next term from the dividend and append it to the difference you just calculated.

        divisor
____________________
quotient term1
dividend - (divisor * quotient term1)
                dividend term2

Step 5: Repeat

Repeat steps 2 to 4 until you've brought down all the terms from the dividend.

Continue dividing, multiplying, and subtracting until you can't bring down any more terms. The terms that remain become the remainder.

Step 6: Write the Result

The quotient is the sum of the terms you found during the division process. The remainder, if any, is written as a fraction with the divisor as the denominator.

The final result is the quotient plus the remainder (if any).

Here's a step-by-step example:

Divide ${�}^3-2{�}^2+3�-5$ by $�-1$.

1. Arrange the polynomials:

   
            x - 1
   ____________________
   x^3 - 2x^2 + 3x - 5
   

2. Divide the first term:

   • Divide ${�}^3$ by $�$, which gives ${�}^2$ as the first term of the quotient.

   
            x - 1
   ____________________
   x^3 - 2x^2 + 3x - 5
   x^2
   

3. Multiply and subtract:

   • Multiply $(�-1)$ by ${�}^2$, which gives ${�}^3-{�}^2$.
   • Subtract this from the dividend.

   
            x - 1
   ____________________
   x^3 - 2x^2 + 3x - 5
   x^2 - (x^3 - x^2) = x^2 + x^2 = 2x^2
   

4. Bring down the next term:

   • Bring down $3�$.

   
            x - 1
   ____________________
   x^3 - 2x^2 + 3x - 5
   x^2 + 2x^2
   3x
   

5. Repeat steps 2 to 4 for the next term:

   • Divide $3�$ by $�$, which gives $3$ as the next term of the quotient.
   • Multiply $(�-1)$ by $3$, which gives $3�-3$.
   • Subtract this from the remaining part of the dividend.

   
            x - 1
   ____________________
   x^3 - 2x^2 + 3x - 5
   x^2 + 2x^2
   3x - (3x - 3) = 3
   

6. No more terms can be brought down, so we have our result:

• The quotient is ${�}^2+3$.
• The remainder is $3$.
• So, ${�}^3-2{�}^2+3�-5$ divided by $�-1$ equals ${�}^2+3$ with a remainder of $3$.

Here are a couple more examples of dividing polynomials using long division:

Example 1: Divide $2{�}^3-5{�}^2+3�-7$ by $�-2$.

1. Set up the division:

          2x^2 + 3
       _________________________
x - 2 | 2x^3 - 5x^2 + 3x - 7

2. Divide the first term:

Divide $2{�}^3$ by $�$, which equals $2{�}^2$.

          2x^2 + 3
       _________________________
x - 2 | 2x^3 - 5x^2 + 3x - 7
       2x^2

3. Multiply and subtract:

• Multiply $(�-2)$ by $2{�}^2$, which equals $2{�}^3-4{�}^2$.
• Subtract this result from $2{�}^3-5{�}^2$, which equals ${�}^2$.

          2x^2 + 3
       _________________________
x - 2 | 2x^3 - 5x^2 + 3x - 7
       2x^2 - (2x^3 - 4x^2)
                  x^2

4. Bring down the next term:

Bring down $+3�$.

          2x^2 + 3
       _________________________
x - 2 | 2x^3 - 5x^2 + 3x - 7
       2x^2 - (2x^3 - 4x^2)
                  x^2
                  + 3x

5. Repeat:

Divide ${�}^2$ by $�$, which equals $�$.

Multiply $(�-2)$ by $�$, which equals ${�}^2-2�$.

Subtract ${�}^2-2�$ from ${�}^2$, which equals $3�$.

          2x^2 + 3
       _________________________
x - 2 | 2x^3 - 5x^2 + 3x - 7
       2x^2 - (2x^3 - 4x^2)
                  x^2
                  + 3x
                  - (x^2 - 2x)
                   3x

6. No more terms can be brought down, so we have our result:

The quotient is $2{�}^2+�+3$.

There's no remainder in this case.

So, $2{�}^3-5{�}^2+3�-7$ divided by $�-2$ equals $2{�}^2+�+3$.

---

Example 2: Divide $3{�}^4+2{�}^3-5{�}^2+4�-8$ by ${�}^2-2�+2$.

1. Set up the division:

           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8

2. Divide the first term:

Divide $3{�}^4$ by ${�}^2$, which equals $3{�}^2$.

           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8
       3x^2

3. Multiply and subtract:

• Multiply $({�}^2-2�+2)$ by $3{�}^2$, which equals $3{�}^4-6{�}^3+6{�}^2$.
• Subtract this result from $3{�}^4+2{�}^3$, which equals $6{�}^3-6{�}^2$.

           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8
       3x^2 - (3x^4 - 6x^3 + 6x^2)
               6x^3 - 9x^2

4. Bring down the next term:

Bring down $+4�$.

           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8
       3x^2 - (3x^4 - 6x^3 + 6x^2)
               6x^3 - 9x^2
                       + 4x

5. Repeat:

Divide $6{�}^3$ by ${�}^2$, which equals $6�$.

Multiply $({�}^2-2�+2)$ by $6�$, which equals $6{�}^3-12{�}^2+12�$.

Subtract $6{�}^3-12{�}^2+12�$ from $6{�}^3-9{�}^2$, which equals $3{�}^2+12�$.

           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8
       3x^2 - (3x^4 - 6x^3 + 6x^2)
               6x^3 - 9x^2
                       + 4x
                       - (3x^2 + 12x)
                           -9x

6. Bring down the next term:

Bring down $-8$.

           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8
       3x^2 - (3x^4 - 6x^3 + 6x^2)
               6x^3 - 9x^2
                       + 4x
                       - (3x^2 + 12x)
                           -9x
                              - 8

7. Repeat:

Divide $-9�$ by ${�}^2$, which equals $-9�$.

Multiply $({�}^2-2�+2)$ by $-9�$, which equals $-9{�}^3+18{�}^2-18�$.

Subtract $-9{�}^3+18{�}^2-18�$ from $-9�$, which equals $0$.

scss
           3x^2 + 2
       _________________________
x^2 - 2x + 2 | 3x^4 + 2x^3 - 5x^2 + 4x - 8
       3x^2 - (3x^4 - 6x^3 + 6x^2)
               6x^3 - 9x^2
                       + 4x
                       - (3x^2 + 12x)
                           -9x
                              - 8
                            - (0)

8. No more terms can be brought down, so we have our result:

The quotient is $3{�}^2+6�+9$.

There's no remainder in this case.

So, $3{�}^4+2{�}^3-5{�}^2+4�-8$ divided by ${�}^2-2�+2$ equals $3{�}^2+6�+9$.

The Division Algorithm is a fundamental concept in mathematics that provides a way to divide one integer (the dividend) by another integer (the divisor) and obtain both a quotient and a remainder. It's a generalization of the long division method for integers. The Division Algorithm can be applied to various mathematical structures, including integers, polynomials, and more.

Here's a general description of the Division Algorithm for integers:

Division Algorithm for Integers:

Given two integers, $�$ (the dividend) and $�$ (the divisor), where $�$ is not equal to zero, the Division Algorithm states that there exist unique integers $�$ (the quotient) and $�$ (the remainder) such that:

$�=��+�$

• $�$ is the quotient, and it represents how many times $�$ can be subtracted from $�$ without going below zero.
• $�$ is the remainder, and it is the positive integer that remains when $�$ is divided by $�$.

Furthermore, the remainder $�$ must satisfy the inequality $0\le �<\mathrm{\mid }�\mathrm{\mid }$.

In the case of polynomials, the Division Algorithm is applied similarly, but instead of integers, you divide one polynomial by another, obtaining a polynomial quotient and a polynomial remainder.

The Division Algorithm is a fundamental tool in various areas of mathematics, including number theory, algebra, and computer science. It plays a crucial role in solving problems related to divisibility, factorization, and finding solutions to equations.

For example, when performing long division of integers or dividing polynomials, you're essentially applying the Division Algorithm. The same principles apply in modular arithmetic when finding remainders after division.

In summary, the Division Algorithm is a foundational concept that provides a systematic way to divide one quantity by another, producing both a quotient and a remainder, and it has applications across different branches of mathematics.

Synthetic division is a quick and efficient method for dividing a polynomial by a linear divisor of the form $�-�$, where $�$ is a constant. This method is particularly useful for finding roots (zeros) of a polynomial and simplifying the division process. Here's how to use synthetic division:

Step 1: Set Up the Division

1. Write down the coefficients of the polynomial in descending order of their degrees, including any missing terms with a coefficient of zero. The polynomial should be in the form $�{�}^{�}+�{�}^{�-1}+\dots +�$, where $�,�,\dots ,�$ are the coefficients.

2. Write down the divisor of the form $�-�$, where $�$ is a constant. Make sure the divisor is in the form $�-�$, not $�+�$. Synthetic division only works for divisors in this form.

Step 2: Perform Synthetic Division

1. Bring down the first coefficient of the polynomial (the leading coefficient) into the division bar.

2. Multiply the number in the division bar by the constant $�$ from the divisor and write the result in the next row below the polynomial.

3. Add the values in the first column of the division bar (the result from step 2) to the second coefficient of the polynomial. Write this result in the next row.

4. Repeat steps 2 and 3 for each subsequent row, moving from left to right, until you reach the last coefficient of the polynomial.

Step 3: Interpret the Result

• The last row of values obtained after performing synthetic division represents the coefficients of the quotient polynomial. The last value in this row is the remainder.

• The quotient polynomial is formed by the coefficients in the last row, starting from the left and proceeding to the right.

Example:

Divide $3{�}^3-5{�}^2+2�-8$ by $�-2$.

1. Set Up the Division:

   
   Polynomial:   3    -5    2    -8
   Divisor:      2
   

2. Perform Synthetic Division:

   
                3    -5    2    -8
   --------------------------------
   2 |  3    -5    2    -8
        6     2    8
   --------------------------------
                9    -3    10
   

3. Interpret the Result:

   • The last row represents the coefficients of the quotient polynomial: $3{�}^2-3�+10$.
   • The remainder is 10.

So, $3{�}^3-5{�}^2+2�-8$ divided by $�-2$ equals $3{�}^2-3�+10$ with a remainder of 10.

This method is particularly useful when dividing polynomials by linear factors, such as finding roots or simplifying expressions.

Polynomial division can be a useful tool for solving application problems that involve finding roots (zeros) of a polynomial equation or determining the solution to a real-world problem. Here's a general process for using polynomial division to solve application problems:

Step 1: Formulate the Problem

First, understand the nature of the problem and translate it into a mathematical equation. This often involves creating a polynomial equation that represents the situation.

Step 2: Set Up the Equation

Write down the polynomial equation that represents the problem. This equation may involve variables, constants, and a polynomial expression.

Step 3: Use Polynomial Division

If the problem involves finding roots or solving for a specific variable, use polynomial division to simplify the equation. This typically means dividing the polynomial equation by another polynomial or a linear binomial to find the roots or simplify the equation.

Step 4: Solve for Variables

Once you've simplified the equation using polynomial division, you can often solve for the variables involved in the problem. This may require further algebraic manipulation or factoring.

Step 5: Interpret the Solution

Interpret the mathematical solution in the context of the original problem. What do the values of the variables represent, and what is their significance in the real-world situation?

Step 6: Check Your Solution

Always verify your solution to ensure it makes sense in the context of the problem. Check if the values you found satisfy the original problem's conditions and constraints.

Step 7: Communicate Your Answer

Present your solution clearly, both mathematically and in plain language, so that anyone reading your solution can understand your reasoning and the implications of your findings.

Here's a simple example:

Problem: Suppose you have a rectangular garden with a length of $2{�}^2+3�$ meters and a width of $�$ meters. If the area of the garden is 30 square meters, find the value of $�$.

Step 1: Formulate the Problem We have a rectangular garden with known dimensions and an area. We want to find the value of $�$, which represents the width of the garden.

Step 2: Set Up the Equation The area of a rectangle is given by length × width. So, the equation representing the problem is: $(2{�}^2+3�)\cdot �=30$

Step 3: Use Polynomial Division We can simplify this equation by multiplying out the terms: $2{�}^3+3{�}^2=30$

Step 4: Solve for Variables Now, we have a polynomial equation that we can solve using polynomial division. We need to find the roots (zeros) of the polynomial. In this case, we can factor it: ${�}^2(2�+3)=30$

Now, set each factor equal to zero and solve for $�$: ${�}^2=0⟹�=0$ $2�+3=0⟹2�=-3⟹�=-\frac{3}{2}$

Step 5: Interpret the Solution In the context of the problem, $�=0$ doesn't make sense because it would imply a garden with no width. Therefore, $�=-\frac{3}{2}$ is the valid solution. The width of the garden is $-\frac{3}{2}$ meters, which is not practical, so there may be an issue with the problem statement.

Step 6: Check Your Solution Verify that the width of the garden, $-\frac{3}{2}$ meters, is a reasonable answer in the context of the problem. In this case, it doesn't make sense, so you might want to double-check the problem statement.

Step 7: Communicate Your Answer Clearly state your findings, including the value of $�$ and any issues or concerns you have with the problem.

Polynomial division can be a valuable tool for solving a wide range of application problems in algebra and calculus. It allows you to simplify complex equations and find solutions to real-world situations.

Here are a couple of examples that demonstrate how to use polynomial division to solve application problems:

Example 1: Finding the Roots of a Polynomial Equation

Problem: Solve for $�$ in the equation $2{�}^3-5{�}^2-4�+8=0$.

Step 1: Formulate the Problem We want to find the values of $�$ that satisfy the given polynomial equation.

Step 2: Set Up the Equation The equation is already given: $2{�}^3-5{�}^2-4�+8=0$.

Step 3: Use Polynomial Division In this case, we don't need polynomial division since the equation is already in standard form.

Step 4: Solve for Variables We can use methods like factoring, the Rational Root Theorem, or numerical methods to find the roots of the polynomial.

By using the Rational Root Theorem, we can determine that $�=2$ is one of the roots. We can then use synthetic division to find the remaining quadratic factor: $(2�-1)(�-4)=0$.

So, the solutions are $�=2$, $�=\frac{1}{2}$, and $�=4$.

Step 5: Interpret the Solution The solutions represent the values of $�$ that make the polynomial equation true. In this context, they are the roots of the polynomial.

Step 6: Check Your Solution Verify that substituting the solutions back into the original equation indeed makes it true.

Step 7: Communicate Your Answer Clearly state the values of $�$ that satisfy the equation.

---

Example 2: Revenue Maximization

Problem: A company's revenue, in dollars, is modeled by the polynomial $�(�)=-3{�}^2+30�$, where $�$ is the number of items sold. Find the value of $�$ that maximizes the company's revenue.

Step 1: Formulate the Problem We want to find the value of $�$ that maximizes the company's revenue, which is modeled by the given polynomial.

Step 2: Set Up the Equation The equation representing the problem is the revenue function: $�(�)=-3{�}^2+30�$.

Step 3: Use Polynomial Division No polynomial division is needed for this problem.

Step 4: Solve for Variables To maximize revenue, we need to find the vertex of the parabolic revenue function. The vertex occurs at the $�$-coordinate $-\frac{�}{2�}$, where $�$ and $�$ are coefficients from the standard form $�{�}^2+��+�$.

In this case, $�=-3$ and $�=30$, so the $�$-coordinate of the vertex is $�=-\frac{30}{2(-3)}=5$.

Step 5: Interpret the Solution The value of $�=5$ represents the number of items sold that maximizes the company's revenue.

Step 6: Check Your Solution Check that the vertex occurs at $�=5$ by verifying the vertex form of the parabola.

Step 7: Communicate Your Answer State that the value of $�=5$ maximizes the company's revenue.

These examples demonstrate how polynomial division can be used to solve a variety of application problems, from finding roots of polynomial equations to optimizing real-world scenarios.