Expressing the Quotient and Remainder

Oct 13, 2025 by miodeya.com 57 views

Introduction

In algebra, division of polynomials is a crucial operation that helps us simplify complex expressions and solve equations. When dividing a polynomial by another, we aim to find the quotient and remainder. In this article, we will focus on dividing the expression $$2y^2 + 8$$ by $$2y + 4$$ and determine which expression represents the quotient and remainder.

The Division Process

To divide $$2y^2 + 8$$ by $$2y + 4$$, we will use long division, a method that involves dividing the highest degree term of the dividend by the highest degree term of the divisor. The process involves several steps:

Divide the leading term of the dividend ($$2y^2$$) by the leading term of the divisor ($$2y$$).
Multiply the entire divisor by the result from step 1 and subtract the product from the dividend.
Bring down the next term of the dividend and repeat the process until we have no more terms to bring down.

Step 1: Divide the Leading Term

We start by dividing the leading term of the dividend ($$2y^2$$) by the leading term of the divisor ($$2y$$). This gives us $$y$$.

Step 2: Multiply and Subtract

Next, we multiply the entire divisor ($$2y + 4$$) by the result from step 1 ($$y$$). This gives us $$2y^2 + 4y$$. We then subtract this product from the dividend ($$2y^2 + 8$$). This leaves us with a remainder of $$4y + 8$$.

Step 3: Bring Down the Next Term

Since we have no more terms to bring down, we can stop here. The remainder is $$4y + 8$$.

Step 4: Write the Quotient and Remainder

The quotient is the result from step 1, which is $$y$$. The remainder is $$4y + 8$$.

Expressing the Quotient and Remainder

We can express the quotient and remainder as a single expression. We can write the quotient as $$y$$ and the remainder as $$4y + 8$$. However, we can also express the remainder as a fraction with the divisor as the denominator. This gives us $$\frac{4y + 8}{2y + 4}$$.

Simplifying the Remainder

We can simplify the remainder by factoring out the greatest common factor (GCF) of the numerator and denominator. The GCF of $$4y + 8$$ and $$2y + 4$$ is 4. Factoring out 4 gives us $$\frac{4(y + 2)}{2(y + 2)}$$.

Canceling Common Factors

We can cancel out the common factors in the numerator and denominator. The common factor is $$(y + 2)$$. Canceling this factor gives us $$\frac{4}{2}$$.

Simplifying the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). The GCF of 4 and 2 is 2. Dividing both the numerator and denominator by 2 gives us $$\frac{2}{1}$$, which simplifies to 2.

Writing the Final Expression

We can now write the final expression for the quotient and remainder. The quotient is $$y$$ and the remainder is $$2$$.

Conclusion

In conclusion, when dividing $$2y^2 + 8$$ by $$2y + 4$$, the quotient is $$y$$ and the remainder is $$2$$. This can be expressed as the expression $$y - 2 + \frac{16}{2y + 4}$$.

Final Answer

The final answer is: $\boxed{y - 2 + \frac{16}{2y + 4}}$

Introduction

In our previous article, we explored the process of dividing the expression $$2y^2 + 8$$ by $$2y + 4$$ and determined that the quotient is $$y$$ and the remainder is $$2$$. However, we also saw that the remainder can be expressed as a fraction with the divisor as the denominator. In this article, we will answer some frequently asked questions about the division process and provide additional insights into the quotient and remainder.

Q&A

Q: What is the quotient when dividing $$2y^2 + 8$$ by $$2y + 4$$?

A: The quotient is $$y$$.

Q: What is the remainder when dividing $$2y^2 + 8$$ by $$2y + 4$$?

A: The remainder is $$2$$.

Q: Can the remainder be expressed as a fraction with the divisor as the denominator?

A: Yes, the remainder can be expressed as a fraction with the divisor as the denominator. In this case, the remainder is $$\frac{16}{2y + 4}$$.

Q: How do I simplify the remainder as a fraction?

A: To simplify the remainder as a fraction, you can factor out the greatest common factor (GCF) of the numerator and denominator. In this case, the GCF is 4. Factoring out 4 gives you $$\frac{4(y + 2)}{2(y + 2)}$$.

Q: Can I cancel out common factors in the numerator and denominator?

A: Yes, you can cancel out common factors in the numerator and denominator. In this case, the common factor is $$(y + 2)$$. Canceling this factor gives you $$\frac{4}{2}$$.

Q: How do I simplify the fraction further?

A: To simplify the fraction further, you can divide the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 2. Dividing both the numerator and denominator by 2 gives you $$\frac{2}{1}$$, which simplifies to 2.

Q: What is the final expression for the quotient and remainder?

A: The final expression for the quotient and remainder is $$y - 2 + \frac{16}{2y + 4}$$.

Additional Insights

The quotient and remainder are two separate components of the division process.
The quotient is the result of dividing the highest degree term of the dividend by the highest degree term of the divisor.
The remainder is the result of subtracting the product of the divisor and the quotient from the dividend.
The remainder can be expressed as a fraction with the divisor as the denominator.
The fraction can be simplified by factoring out the greatest common factor (GCF) of the numerator and denominator.

Conclusion

In conclusion, dividing $$2y^2 + 8$$ by $$2y + 4$$ results in a quotient of $$y$$ and a remainder of $$2$$. The remainder can be expressed as a fraction with the divisor as the denominator, and can be simplified by factoring out the greatest common factor (GCF) of the numerator and denominator. We hope this Q&A guide has provided additional insights into the division process and has helped you better understand the quotient and remainder.

Final Answer

The final answer is: $\boxed{y - 2 + \frac{16}{2y + 4}}$