# Dividing Polynomials Worksheet, Steps, and Examples

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### Key Points about Dividing Polynomials

- Dividing polynomials involves dividing one polynomial by another to obtain a quotient and remainder.
- There are several methods for dividing polynomials, including long division, synthetic division, and the box method.
- Dividing polynomials is an essential skill that is necessary for solving algebraic equations and graphing polynomial functions.

**Dividing Polynomials: The Complete Guide**

Dividing polynomials is a fundamental concept in algebra. It involves the process of dividing one polynomial by another, resulting in a quotient and remainder. Polynomials are expressions that consist of variables, coefficients, and exponents, and dividing them requires a solid understanding of algebraic operations.

There are several methods for dividing polynomials, including long division, synthetic division, and the box method. Long division is the most commonly used method, and it involves dividing the polynomial term by term. Synthetic division is a quicker method that is used to divide a polynomial by a linear factor. The box method is a visual method that is used to divide polynomials with two or more terms.

Dividing polynomials is an essential skill that is used in various fields, including engineering, physics, and computer science. It is necessary for solving algebraic equations and graphing polynomial functions. Understanding how to divide polynomials is crucial for students who want to excel in algebra and beyond.

**Common Core Standard:**

**How to Divide Polynomials**

Dividing polynomials is an important mathematical operation that is used in various fields, including physics, engineering, and economics. In this section, we will discuss two methods of dividing polynomials: the long division method and the synthetic division method. Additionally, we will also discuss how to solve for X.

**Long Division Method**

The long division method is a standard procedure for dividing polynomials. It is similar to the long division method used for dividing whole numbers. The method involves dividing the polynomial into smaller parts until it can no longer be divided. Here are the steps to follow when using the long division method:

- Write the dividend (the polynomial being divided) and the divisor (the polynomial dividing the dividend) in long division format.
- Divide the first term of the dividend by the first term of the divisor. Write the answer above the dividend.
- Multiply the divisor by the answer obtained in step 2. Write the result below the dividend.
- Subtract the result obtained in step 3 from the dividend. Write the remainder below the line.
- Bring down the next term of the dividend and place it next to the remainder obtained in step 4.
- Repeat steps 2 to 5 until there are no more terms in the dividend.

**Synthetic Division Method**

The synthetic division method is a shorthand method for dividing polynomials. It is used when the divisor is a linear polynomial of the form (x – a). Here are the steps to follow when using the synthetic division method:

- Write the coefficients of the dividend in descending order.
- Write the value of a next to the coefficients.
- Draw a line below the coefficients and a box to the left of the line.
- Bring down the first coefficient into the box.
- Multiply the value of a by the number in the box and write the result below the second coefficient.
- Add the two numbers obtained in step 4 and write the result below the third coefficient.
- Repeat steps 5 and 6 until all the coefficients have been used.
- The answer is the coefficients of the quotient, excluding the last number.

**Solving for X**

To solve for X, you need to isolate X on one side of the equation. Here are the steps to follow when solving for X:

- Simplify the polynomial equation by combining like terms.
- Move all the terms to one side of the equation.
- Factor out any common factors.
- Use the zero product property to solve for X.
- Check your answer by substituting the value of X back into the original equation.

In conclusion, dividing polynomials is an essential mathematical skill that is used in many fields. The long division method and the synthetic division method are two effective ways to divide polynomials. Additionally, solving for X requires isolating X on one side of the equation and using the zero product property.

**Dividing Polynomials Using Long Division**

Long division is a method of dividing polynomials that involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by the quotient term, subtracting the result from the dividend, bringing down the next term of the dividend, and repeating the process until there is a remainder of lower degree than the divisor. The process is similar to long division in arithmetic.

To illustrate how to divide polynomials using long division, consider the following example:

Divide x^3 + 2x^2 – 5x – 6 by x + 3

First, write the dividend and divisor in long division format:

x^2 – x – 6

x + 3 | x^3 + 2x^2 – 5x – 6

Next, divide the leading term of the dividend, x^3, by the leading term of the divisor, x, to get the quotient term, x^2. Multiply the divisor, x + 3, by the quotient term, x^2, to get x^3 + 3x^2. Subtract this result from the dividend to get the first remainder, 2x^2 – 5x – 6.

x^2 – x – 6

x + 3 | x^3 + 2x^2 – 5x – 6

– (x^3 + 3x^2)

————–

– x^2 – 5x

Bring down the next term of the dividend, -5x, and repeat the process. Divide the leading term of the new dividend, -x^2, by the leading term of the divisor, x, to get the quotient term, -x. Multiply the divisor, x + 3, by the quotient term, -x, to get -x^2 – 3x. Subtract this result from the dividend to get the second remainder, -2x – 6.

x^2 – x – 6

x + 3 | x^3 + 2x^2 – 5x – 6

– (x^3 + 3x^2)

————–

– x^2 – 5x

– (-x^2 – 3x)

————

-2x – 6

Finally, bring down the last term of the dividend, -6, and repeat the process. Divide the leading term of the new dividend, -2x, by the leading term of the divisor, x, to get the quotient term, -2. Multiply the divisor, x + 3, by the quotient term, -2, to get -2x – 6. Subtract this result from the dividend to get the final remainder, 0.

x^2 – x – 6

x + 3 | x^3 + 2x^2 – 5x – 6

– (x^3 + 3x^2)

————–

– x^2 – 5x

– (-x^2 – 3x)

————

-2x – 6

– (-2x – 6)

———

0

Therefore, the quotient is x^2 – x – 2 and the remainder is 0.

Long division of polynomials can be a useful method for dividing polynomials of any degree. It is important to note that not all polynomials can be divided using long division. In such cases, other methods such as synthetic division may be used.

**Synthentic Division of Polynomials**

Synthetic division is a shorthand method of dividing a polynomial by a linear factor of the form (x-a). It is a quick and easy way to divide polynomials, especially when the divisor is a linear factor. The method is based on the fact that if a polynomial is divided by (x-a), then the remainder is the value of the polynomial evaluated at a.

The synthetic division process involves writing the coefficients of the polynomial in a row and performing a series of simple arithmetic operations. The value a is then used to divide the first coefficient, and the result is written below the second coefficient. This process is repeated until all the coefficients have been divided.

One of the advantages of synthetic division is that it can be used to find the roots of a polynomial. If the remainder is zero, then the value of a is a root of the polynomial. Synthetic division can also be used to factorize polynomials. If the result of the division is a polynomial of degree one less than the original polynomial, then the quotient is a factor of the original polynomial.

Synthetic division can be performed using a table, which makes the process easier to follow. Here is an example of how to use synthetic division to divide the polynomial 2x^3 + 3x^2 – 5x – 6 by (x-2):

| 2 | 3 | -5 | -6 | | – | 4 | 14 | 18 | | 2 | 7 | 9 | 12 |

The first row of the table contains the coefficients of the polynomial, and the second row contains the value of a (in this case, 2) multiplied by the coefficients. The third row contains the result of subtracting the second row from the first row. The final result is the polynomial 2x^2 + 7x + 9 with a remainder of 12.

Overall, synthetic division is a useful tool for dividing polynomials, finding roots, and factorizing polynomials. It is a quick and easy method that can save time and effort when working with polynomials.

**Dividing Polynomials Examples**

Dividing polynomials can be a challenging task, but with practice, it becomes easier. Here are a few examples to help understand the process better:

**Example 1:**

Divide 5x^3 – 3x^2 + 2x – 1 by x – 2.

First, write the polynomial division in long division format:

5x^2 + 7x + 12

x – 2 | 5x^3 – 3x^2 + 2x – 1

-5x^3 + 10x^2

—————

7x^2 + 2x – 1

-7x^2 + 14x

————

16x – 1

Therefore, 5x^3 – 3x^2 + 2x – 1 divided by x – 2 equals 5x^2 + 7x + 12 with a remainder of 16x – 1.

**Example 2:**

Divide 3x^4 + 2x^3 – 5x^2 – x + 2 by x + 2.

First, write the polynomial division in long division format:

3x^3 – 4x^2 + 3x – 7

x + 2 | 3x^4 + 2x^3 – 5x^2 – x + 2

-3x^4 – 6x^3

—————

-4x^3 – 5x^2

4x^3 + 8x^2

————

3x^2 – x

-3x^2 – 6x

———–

5x + 2

Therefore, 3x^4 + 2x^3 – 5x^2 – x + 2 divided by x + 2 equals 3x^3 – 4x^2 + 3x – 7 with a remainder of 5x + 2.

**Example 3:**

Divide 6x^3 + 5x^2 – 3x + 1 by 2x – 1.

First, write the polynomial division in long division format:

3x^2 + 7x + 10

2x – 1 | 6x^3 + 5x^2 – 3x + 1

-6x^3 + 3x^2

—————

8x^2 – 3x + 1

-8x^2 + 4x

————

x + 1

Therefore, 6x^3 + 5x^2 – 3x + 1 divided by 2x – 1 equals 3x^2 + 7x + 10 with a remainder of x + 1.

These examples demonstrate how to divide polynomials using long division format. By following the steps in these examples, one can become proficient in dividing polynomials.

**Polynomial Long Division Examples**

Polynomial long division is a method used to divide polynomials. It is a bit similar to the long division method used to divide numbers. In this method, the polynomial to be divided is written in standard form, and the divisor is written in the same way. Then, the division process is carried out, just like long division. Here are a few examples of polynomial long division:

**Example 1**

Divide x^3 + 2x^2 – 4x – 8 by x – 2.

First, set up the division problem like this:

x^2 + 4x + 4

___________________

x – 2 | x^3 + 2x^2 – 4x – 8

Then, divide the first term of the dividend (x^3) by the first term of the divisor (x). This gives x^2. Write this above the line. Multiply the divisor (x – 2) by x^2 to get x^3 – 2x^2. Write this below the dividend, and subtract it from the dividend to get:

x^2 + 4x + 4

___________________

x – 2 | x^3 + 2x^2 – 4x – 8

– (x^3 – 2x^2)

_____________

4x^2 – 4x

Now, bring down the next term of the dividend (-4x) and repeat the process:

x^2 + 4x + 4

___________________

x – 2 | x^3 + 2x^2 – 4x – 8

– (x^3 – 2x^2)

_____________

4x^2 – 4x

– (4x^2 – 8x)

____________

4x – 8

Finally, bring down the last term of the dividend (-8) and repeat the process:

x^2 + 4x + 4

___________________

x – 2 | x^3 + 2x^2 – 4x – 8

– (x^3 – 2x^2)

_____________

4x^2 – 4x

– (4x^2 – 8x)

____________

4x – 8

– (4x – 8)

________

0

The remainder is zero, which means that x^3 + 2x^2 – 4x – 8 is divisible by x – 2.

**Example 2**

Divide 3x^3 – 5x^2 + 2x + 1 by x – 1.

First, set up the division problem like this:

3x^2 – 2x – 4

___________________

x – 1 | 3x^3 – 5x^2 + 2x + 1

Then, divide the first term of the dividend (3x^3) by the first term of the divisor (x). This gives 3x^2. Write this above the line. Multiply the divisor (x – 1) by 3x^2 to get 3x^3 – 3x^2. Write this below the dividend, and subtract it from the dividend to get:

3x^2 – 2x – 4

___________________

x – 1 | 3x^3 – 5x^2 + 2x + 1

– (3x^3 – 3x^2)

_____________

-2x^2 + 2x

Now, bring down the next term of the dividend (2x) and repeat the process:

3x^2 – 2x – 4

___________________

x – 1 | 3x^3 – 5x^2 + 2x + 1

– (3x^3 – 3x^2)

_____________

-2x^2 + 2x

– (-2x^2 + 2x)

_____________

0

The remainder is zero, which means that 3x^3 – 5x^2 + 2x + 1 is divisible by x – 1.

**Dividing Polynomials by Monomials**

Dividing polynomials by monomials is a common operation in algebra. There are two methods that can be used to divide polynomials by monomials: splitting the terms of the polynomial and solving each term separately, or using the factorization method.

**Splitting the Terms of the Polynomial**

The most common method that is used to divide polynomials by monomials is by splitting the terms of the polynomial separated by (+) or (-) and solving each term separately. The final result will be the combination of all the individual results obtained. The steps for dividing a polynomial by a monomial using this method are as follows:

- Divide each term of the polynomial by the monomial.
- Simplify each term by canceling out any common factors.
- Add or subtract the simplified terms to get the final result.

For example, consider the polynomial 5x^3 – 10x^2 + 15x divided by the monomial 5x. Dividing each term of the polynomial by 5x, we get:

5x^3 ÷ 5x = x^2 -10x^2 ÷ 5x = -2x 15x ÷ 5x = 3

Simplifying each term by canceling out any common factors, we get:

x^2 – 2x + 3

Therefore, 5x^3 – 10x^2 + 15x divided by 5x is equal to x^2 – 2x + 3.

**Factorization Method**

Another method of dividing polynomials by monomials is the factorization method. This method involves factoring out the monomial from the polynomial and then simplifying the resulting expression. The steps for dividing a polynomial by a monomial using this method are as follows:

- Factor out the monomial from the polynomial.
- Simplify the resulting expression by canceling out any common factors.

For example, consider the polynomial 2x^2 + 4x divided by the monomial 2x. Factoring out 2x from the polynomial, we get:

2x^2 + 4x = 2x(x + 2)

Simplifying the resulting expression by canceling out any common factors, we get:

x + 2

Therefore, 2x^2 + 4x divided by 2x is equal to x + 2.

Dividing polynomials by monomials is an important skill in algebra and is used in many applications of mathematics. By understanding the methods of dividing polynomials by monomials, one can solve complex algebraic expressions with ease.

**Dividing Polynomials by Binomials**

Dividing polynomials by binomials is a fundamental concept in algebra. It is used to simplify complex expressions and solve equations. The process of dividing polynomials by binomials involves using long division, which may seem complicated at first. However, with practice, it becomes easier to understand and apply.

To divide a polynomial by a binomial, the following steps must be followed:

- Divide the first term of the polynomial by the first term of the binomial.
- Multiply the result obtained in step 1 by the binomial.
- Subtract the result obtained in step 2 from the polynomial.
- Bring down the next term of the polynomial.
- Repeat steps 1-4 until all terms of the polynomial have been divided.

The following example illustrates the process of dividing a polynomial by a binomial:

Divide x^3 + 2x^2 – 5x – 6 by x + 2

x^2 – 2x – 9

x + 2 | x^3 + 2x^2 – 5x – 6

– (x^3 + 2x^2)

————-

– 5x

– ( – 2x – 4)

———

– x – 6

– ( – x – 2)

———

– 4

Therefore, the quotient is x^2 – 2x – 9 and the remainder is -4.

It is important to note that if the degree of the polynomial is less than the degree of the binomial, then the result of the division will be a fraction. In such cases, the polynomial cannot be divided by the binomial.

In summary, dividing polynomials by binomials is a crucial skill in algebra. By following the steps outlined above, it is possible to simplify complex expressions and solve equations.

**Dividing Polynomials Box Method**

Dividing polynomials can be a daunting task for many students. However, the box method is a simple and effective technique that can make the process much easier. This method involves creating a box with the divisor on the left side and the dividend on the top. The resulting table will have a diagonal line that separates the box into four parts.

To use the box method, follow these steps:

- Write the divisor on the left side of the box and the dividend on the top.
- Divide the first term of the dividend by the first term of the divisor and write the quotient in the top right corner of the box.
- Multiply the divisor by the quotient and write the result beneath the dividend.
- Subtract the result from the dividend and write the remainder in the bottom left corner of the box.
- Bring down the next term of the dividend and write it next to the remainder.
- Repeat steps 2-5 until there are no more terms in the dividend.
- Write the final answer as the sum of the quotients in the top row of the box.

Let’s look at an example to better understand the process. Suppose we want to divide the polynomial x^3 + 3x^2 – 4x – 12 by x + 2 using the box method.

x^2 | x | constant | |

x + 2 | x^3 + 3x^2 | -2x^2 – 4x | -12x – 24 |

x^3 + 2x^2 | x^2 – 4x | ||

5x^2 – 4x | -12x – 24 | ||

5x^2 + 10x | 20x | ||

8x – 24 |

The final answer is x^2 + 5x + 8 with a remainder of -24.

The box method can be a helpful tool for dividing polynomials, but it is important to remember to check your answer using other methods such as long division or synthetic division.

**Dividing Polynomials with Fractions**

Dividing polynomials with fractions is similar to dividing regular polynomials, but with an added step of converting the fraction to a multiplication expression. The process involves finding the reciprocal of the fraction and then multiplying it with the polynomial. Here are the steps to divide polynomial fractions:

- Find the reciprocal of the fraction which appears after the division sign and change it into multiplication sign.
- Multiply the polynomial by the reciprocal of the fraction.
- Simplify the resulting polynomial by combining like terms.

Here’s an example to illustrate the process:

(3x^2 + 4x + 2) / (x + 1/2)

To divide this polynomial fraction, first, find the reciprocal of 1/2:

1 / (1/2) = 2

Then, change the division sign to multiplication and multiply the polynomial by the reciprocal:

(3x^2 + 4x + 2) * 2(x + 1)

Simplify the resulting polynomial by multiplying and combining like terms:

6x^3 + 11x^2 + 9x + 2

Therefore, (3x^2 + 4x + 2) / (x + 1/2) = 6x^3 + 11x^2 + 9x + 2.

It’s important to note that when dividing polynomial fractions, it’s crucial to check for common factors and simplify the polynomial fraction before dividing. This will make the process easier and avoid errors.

**How to Divide Polynomials FAQ**

**How to divide polynomials using synthetic division?**

Synthetic division is a simplified method of dividing polynomials that involves using only the coefficients of the polynomial. It is a quicker and more efficient method than long division. To use synthetic division, the divisor must be in the form of x – a, where a is a constant.

Here are the steps to divide polynomials using synthetic division:

- Write the coefficients of the polynomial in descending order.
- Write the constant term of the divisor next to the coefficients.
- Bring down the first coefficient.
- Multiply the constant term of the divisor by the coefficient you just brought down and write the result below the next coefficient.
- Add the result to the next coefficient.
- Repeat steps 4 and 5 until all coefficients have been used.
- The final row of coefficients represents the quotient.

**What are the 2 methods to divide polynomials?**

There are two main methods to divide polynomials: long division and synthetic division. Long division is a more traditional method that involves dividing each term of the polynomial by the divisor, similar to how you would divide two numbers. Synthetic division, as mentioned earlier, involves only using the coefficients of the polynomial and is faster than long division.

**How do you divide a polynomial?**

To divide a polynomial, you need to follow the steps of either long division or synthetic division. First, determine the divisor and make sure it is in the correct form. Then, follow the steps of the chosen method to divide the polynomial. Remember to simplify the final answer as much as possible by factoring out any common factors.

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