Applying the Long Division Process

Oct 13, 2025 by miodeya.com 86 views

Introduction

Long division is a mathematical technique used to divide polynomials and find the quotient. In this article, we will focus on finding the quotient of a given polynomial using long division and expressing it as a rational expression. We will use the example of dividing the polynomial $2x^2 - 10x + 15$ by $x - 5$ to demonstrate the process.

The Long Division Process

To find the quotient using long division, we need to follow these steps:

Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from step 1.
Subtract the product from step 2 from the dividend.
Bring down the next term of the dividend.
Repeat steps 1-4 until the degree of the remainder is less than the degree of the divisor.

Applying the Long Division Process

Let's apply the long division process to the given example:

${ x - 5 \longdiv {2x^2 - 10x + 15} }$

Step 1: Divide the leading term of the dividend ( $2x^2$ ) by the leading term of the divisor ( $x$ ).

${ \frac{2x^2}{x} = 2x }$

Step 2: Multiply the entire divisor ( $x - 5$ ) by the result from step 1 ( $2x$ ).

${ (x - 5) \times 2x = 2x^2 - 10x }$

Step 3: Subtract the product from step 2 from the dividend.

${ 2x^2 - 10x + 15 - (2x^2 - 10x) = 15 }$

Step 4: Bring down the next term of the dividend.

Since there are no more terms to bring down, we can stop here.

Expressing the Quotient as a Rational Expression

The result from step 1 is the first term of the quotient, which is $2x$ . Since the degree of the remainder ( $15$ ) is less than the degree of the divisor ( $x - 5$ ), we can express the quotient as a rational expression.

The quotient is $2x + \frac{15}{x-5}$ .

However, we can simplify this expression further by combining the terms.

${ 2x + \frac{15}{x-5} = \frac{2x(x-5) + 15}{x-5} }$

${ = \frac{2x^2 - 10x + 15}{x-5} }$

Conclusion

In this article, we used the long division process to find the quotient of the polynomial $2x^2 - 10x + 15$ by $x - 5$ . We then expressed the quotient as a rational expression. The final answer is $\boxed{\frac{2x^2 - 10x + 15}{x-5}}$ .

Answer Key

The correct answer is:

${ \frac{2x^2 - 10x + 15}{x-5} }$

This is option A.

Discussion

Do you have any questions about the long division process or expressing the quotient as a rational expression? Let's discuss in the comments below.

References

Note: The references provided are for educational purposes only and are not affiliated with this article.

Introduction

In our previous article, we discussed how to find the quotient using long division and express it as a rational expression. We used the example of dividing the polynomial $2x^2 - 10x + 15$ by $x - 5$ to demonstrate the process. In this article, we will answer some frequently asked questions about the long division process and expressing the quotient as a rational expression.

Q&A

Q: What is the first step in the long division process?

A: The first step in the long division process is to divide the leading term of the dividend by the leading term of the divisor.

Q: How do I know when to stop the long division process?

A: You can stop the long division process when the degree of the remainder is less than the degree of the divisor.

Q: Can I express the quotient as a rational expression if the degree of the remainder is equal to the degree of the divisor?

A: No, you cannot express the quotient as a rational expression if the degree of the remainder is equal to the degree of the divisor. In this case, the remainder is a polynomial of the same degree as the divisor, and you cannot simplify it further.

Q: How do I simplify the quotient as a rational expression?

A: To simplify the quotient as a rational expression, you can combine the terms in the numerator and denominator.

Q: Can I use the long division process to divide polynomials with complex coefficients?

A: Yes, you can use the long division process to divide polynomials with complex coefficients. The process is the same as for polynomials with real coefficients.

Q: How do I check my work when using the long division process?

A: To check your work when using the long division process, you can multiply the quotient by the divisor and add the remainder. If the result is the original dividend, then your work is correct.

Q: Can I use the long division process to divide polynomials with variables in the denominator?

A: No, you cannot use the long division process to divide polynomials with variables in the denominator. In this case, you need to use a different method, such as factoring or synthetic division.

Conclusion

In this article, we answered some frequently asked questions about the long division process and expressing the quotient as a rational expression. We hope this article has been helpful in clarifying any doubts you may have had about the process.

References

Note: The references provided are for educational purposes only and are not affiliated with this article.

Discussion

Do you have any questions about the long division process or expressing the quotient as a rational expression? Let's discuss in the comments below.

Example Problems

Divide the polynomial $3x^2 + 5x + 2$ by $x + 2$ .
Express the quotient as a rational expression.
Simplify the quotient.

Answer Key

The quotient is $3x + 1$ .
The quotient as a rational expression is $\frac{3x + 1}{x + 2}$ .
The simplified quotient is $\frac{3x + 1}{x + 2}$ .

Note: The example problems and answer key are for educational purposes only and are not affiliated with this article.