Divide The Following Expression. Clearly Write Your Answer As (Quotient) $+\frac{\text{(Remainder)}}{\text{(Divisor)}}$. \left(x^4-2x^2+7x-3\right) \div \left(x^2+x\right ]

Mar 13, 2025 by ADMIN 175 views

Divide the Given Expression: A Step-by-Step Guide

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Introduction

In algebra, division of polynomials is a crucial operation that helps us simplify complex expressions and solve equations. When dividing polynomials, we need to follow a specific procedure to ensure that we obtain the correct quotient and remainder. In this article, we will focus on dividing the given expression $\left(x^4-2x^2+7x-3\right) \div \left(x^2+x\right)$ and clearly write our answer as $(Quotient) + \frac{(Remainder)}{(Divisor)}$ .

Understanding Polynomial Division

Polynomial division is a process of dividing a polynomial by another polynomial. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until we obtain a remainder that is of lower degree than the divisor.

Step 1: Divide the Highest Degree Term

To begin the division process, we need to divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, the highest degree term of the dividend is $x^4$ and the highest degree term of the divisor is $x^2$ . Therefore, we divide $x^4$ by $x^2$ to obtain $x^2$ .

Step 2: Multiply the Divisor by the Result

Next, we multiply the entire divisor by the result obtained in the previous step. In this case, we multiply $x^2+x$ by $x^2$ to obtain $x^4+x^3$ .

Step 3: Subtract the Result from the Dividend

Now, we subtract the result obtained in the previous step from the dividend. In this case, we subtract $x^4+x^3$ from $x^4-2x^2+7x-3$ to obtain $-x^3-2x^2+7x-3$ .

Step 4: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend by the highest degree term of the divisor. In this case, the highest degree term of the new dividend is $-x^3$ and the highest degree term of the divisor is $x^2$ . Therefore, we divide $-x^3$ by $x^2$ to obtain $-x$ .

Step 5: Multiply the Divisor by the Result

Next, we multiply the entire divisor by the result obtained in the previous step. In this case, we multiply $x^2+x$ by $-x$ to obtain $-x^3-x^2$ .

Step 6: Subtract the Result from the Dividend

Now, we subtract the result obtained in the previous step from the new dividend. In this case, we subtract $-x^3-x^2$ from $-x^3-2x^2+7x-3$ to obtain $-x^2+7x-3$ .

Step 7: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend by the highest degree term of the divisor. In this case, the highest degree term of the new dividend is $-x^2$ and the highest degree term of the divisor is $x^2$ . Therefore, we divide $-x^2$ by $x^2$ to obtain $-1$ .

Step 8: Multiply the Divisor by the Result

Next, we multiply the entire divisor by the result obtained in the previous step. In this case, we multiply $x^2+x$ by $-1$ to obtain $-x^2-x$ .

Step 9: Subtract the Result from the Dividend

Now, we subtract the result obtained in the previous step from the new dividend. In this case, we subtract $-x^2-x$ from $-x^2+7x-3$ to obtain $8x-3$ .

Step 10: Write the Quotient and Remainder

Since the degree of the remainder is lower than the degree of the divisor, we can stop the division process. The quotient is $x^2-x$ and the remainder is $8x-3$ .

Conclusion

In conclusion, when dividing the given expression $\left(x^4-2x^2+7x-3\right) \div \left(x^2+x\right)$ , we obtain a quotient of $x^2-x$ and a remainder of $8x-3$ . Therefore, we can write our answer as $(x^2-x) + \frac{(8x-3)}{(x^2+x)}$ .

Final Answer

The final answer is: $\boxed{(x^2-x) + \frac{(8x-3)}{(x^2+x)}}$

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Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial by another polynomial. It is a crucial operation in algebra that helps us simplify complex expressions and solve equations.

Q: Why do we need to divide polynomials?

A: We need to divide polynomials to simplify complex expressions and solve equations. Polynomial division helps us to break down a complex expression into simpler components, making it easier to analyze and solve.

Q: What are the steps involved in polynomial division?

A: The steps involved in polynomial division are:

Divide the highest degree term of the dividend by the highest degree term of the divisor.
Multiply the entire divisor by the result obtained in the previous step.
Subtract the result obtained in the previous step from the dividend.
Repeat the process until we obtain a remainder that is of lower degree than the divisor.

Q: What is the quotient and remainder in polynomial division?

A: The quotient is the result of the division process, while the remainder is the amount left over after the division process is complete.

Q: How do we write the quotient and remainder?

A: We write the quotient and remainder as $(Quotient) + \frac{(Remainder)}{(Divisor)}$ .

Q: What are some common mistakes to avoid in polynomial division?

A: Some common mistakes to avoid in polynomial division include:

Not following the correct order of operations
Not multiplying the entire divisor by the result obtained in the previous step
Not subtracting the result obtained in the previous step from the dividend
Not repeating the process until we obtain a remainder that is of lower degree than the divisor

Q: How do we check our work in polynomial division?

A: We can check our work in polynomial division by multiplying the quotient by the divisor and adding the remainder. If the result is equal to the original dividend, then our work is correct.

Q: What are some real-world applications of polynomial division?

A: Polynomial division has many real-world applications, including:

Simplifying complex expressions in physics and engineering
Solving equations in economics and finance
Modeling population growth and decay in biology
Analyzing data in statistics and data science

Q: Can polynomial division be used to solve quadratic equations?

A: Yes, polynomial division can be used to solve quadratic equations. By dividing a quadratic expression by a linear expression, we can obtain a quotient that is a linear expression and a remainder that is a constant.

Q: Can polynomial division be used to solve cubic equations?

A: Yes, polynomial division can be used to solve cubic equations. By dividing a cubic expression by a quadratic expression, we can obtain a quotient that is a quadratic expression and a remainder that is a linear expression.

Q: Can polynomial division be used to solve higher-degree equations?

A: Yes, polynomial division can be used to solve higher-degree equations. By dividing a polynomial expression by a polynomial expression of lower degree, we can obtain a quotient that is a polynomial expression of lower degree and a remainder that is a polynomial expression of lower degree.

Q: What are some tips for mastering polynomial division?

A: Some tips for mastering polynomial division include:

Practicing regularly to build your skills and confidence
Using visual aids such as diagrams and graphs to help you understand the process
Breaking down complex expressions into simpler components
Checking your work carefully to ensure accuracy

Q: Where can I find more resources on polynomial division?

A: You can find more resources on polynomial division in textbooks, online tutorials, and video lectures. Some popular resources include:

Khan Academy: Polynomial Division
Mathway: Polynomial Division
Wolfram Alpha: Polynomial Division

Q: Can I use polynomial division to solve systems of equations?

A: Yes, polynomial division can be used to solve systems of equations. By dividing a polynomial expression by a polynomial expression of lower degree, we can obtain a quotient that is a polynomial expression of lower degree and a remainder that is a polynomial expression of lower degree. This can help us to solve systems of equations that involve polynomial expressions.