Using Polynomial Long Division

Oct 13, 2025 by talynt.org 13 views

Introduction

In mathematics, division is a fundamental operation that involves finding the quotient of two numbers or expressions. When it comes to dividing polynomials, we need to use the concept of polynomial long division or synthetic division to find the quotient. In this article, we will explore the quotient of $$\left(x^3+8\right) \div(x+2)$$ and provide a step-by-step solution to this problem.

Understanding the Problem

The given problem involves dividing a cubic polynomial $$x^3+8$$ by a linear polynomial $$x+2$$. To find the quotient, we need to use polynomial long division or synthetic division. The quotient will be a polynomial of degree one less than the dividend.

Using Polynomial Long Division

To find the quotient, we can use polynomial long division. The process involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. We repeat this process until we have no more terms to divide.

Step 1: Divide the Leading Term

The leading term of the dividend is $$x^3$$, and the leading term of the divisor is $$x$$. We divide $$x^3$$ by $$x$$ to get $$x^2$$.

Step 2: Multiply the Divisor

We multiply the entire divisor $$x+2$$ by $$x^2$$ to get $$x^3+2x2$$.

Step 3: Subtract the Product

We subtract the product $$x^3+2x2$$ from the dividend $$x^3+8$$ to get $$-2x^2+8$$.

Step 4: Repeat the Process

We repeat the process by dividing the leading term of the result $$-2x^2$$ by the leading term of the divisor $$x$$ to get $$-2x$$. We multiply the entire divisor $$x+2$$ by $$-2x$$ to get $$-2x^2-4x$$. We subtract the product $$-2x^2-4x$$ from the result $$-2x^2+8$$ to get $$4x+8$$.

Step 5: Repeat the Process Again

We repeat the process by dividing the leading term of the result $$4x$$ by the leading term of the divisor $$x$$ to get $$4$$. We multiply the entire divisor $$x+2$$ by $$4$$ to get $$4x+8$$. We subtract the product $$4x+8$$ from the result $$4x+8$$ to get $$0$$.

Conclusion

We have completed the polynomial long division process, and the quotient is $$x^2-2x+4$$. This is the final answer to the problem.

Comparison with Answer Choices

We can compare our answer with the answer choices provided:

A. $$x^2+2x+4$$
B. $$x^2-2x+4$$
C. $$x^2+4$$
D. $$x^2-4$$

Our answer $$x^2-2x+4$$ matches with answer choice B.

Conclusion

In conclusion, the quotient of $$\left(x^3+8\right) \div(x+2)$$ is $$x^2-2x+4$$. We used polynomial long division to find the quotient, and our answer matches with answer choice B.

Final Answer

The final answer is $$\boxed{x^2-2x+4}$$.

Introduction

In our previous article, we explored the quotient of $$\left(x^3+8\right) \div(x+2)$$ and found that the quotient is $$x^2-2x+4$$. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the process of polynomial long division?

A: Polynomial long division is a process of dividing a polynomial by another polynomial. It involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. We repeat this process until we have no more terms to divide.

Q: Why do we need to use polynomial long division?

A: We need to use polynomial long division to find the quotient of two polynomials. It is a fundamental operation in algebra that helps us to simplify complex expressions and solve equations.

Q: What is the difference between polynomial long division and synthetic division?

A: Polynomial long division and synthetic division are two different methods of dividing polynomials. Polynomial long division involves dividing the dividend by the divisor using a step-by-step process, while synthetic division involves using a shortcut method to divide the dividend by the divisor.

Q: Can we use synthetic division to find the quotient of $$\left(x^3+8\right) \div(x+2)$$?

A: Yes, we can use synthetic division to find the quotient of $$\left(x^3+8\right) \div(x+2)$$. However, we need to be careful when using synthetic division, as it can be tricky to apply.

Q: What is the significance of the quotient in this problem?

A: The quotient in this problem is $$x^2-2x+4$$. This quotient represents the result of dividing the dividend $$x^3+8$$ by the divisor $$x+2$$. It is an important concept in algebra that helps us to simplify complex expressions and solve equations.

Q: Can we use the quotient to solve equations?

A: Yes, we can use the quotient to solve equations. For example, if we have an equation like $$x^3+8 = (x+2)(x^2-2x+4)$$, we can use the quotient to simplify the equation and solve for x.

Q: What are some common applications of polynomial long division?

A: Polynomial long division has many common applications in algebra, including:

Simplifying complex expressions
Solving equations
Finding the roots of a polynomial
Factoring polynomials

Conclusion

In conclusion, the quotient of $$\left(x^3+8\right) \div(x+2)$$ is $$x^2-2x+4$$. We used polynomial long division to find the quotient, and our answer matches with answer choice B. We also answered some frequently asked questions related to this problem, including the process of polynomial long division, the difference between polynomial long division and synthetic division, and the significance of the quotient in this problem.

Final Answer

The final answer is $$\boxed{x^2-2x+4}$$.