Use Polynomial Long Division To Find The Quotient And The Remainder When $48x^3 - 274x^2 - 427x - 41$ Is Divided By $x - 7$.Quotient:$\square$\square$\square$\square$Remainder:Write The Answer In

Feb 26, 2025 by ADMIN 199 views

Use Polynomial Long Division to Find the Quotient and Remainder

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Introduction

Polynomial long division is a method used to divide a polynomial by another polynomial of a lower degree. It is a powerful tool in algebra that helps us to simplify complex expressions and find the quotient and remainder when a polynomial is divided by another polynomial. In this article, we will use polynomial long division to find the quotient and remainder when the polynomial $48x^3 - 274x^2 - 427x - 41$ is divided by $x - 7$ .

The Process of Polynomial Long Division

Polynomial long division is similar to the long division method used for numbers. 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 the degree of the remainder is less than the degree of the divisor.

Step 1: Divide the Highest Degree Term

The first step in polynomial long division is 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 $48x^3$ and the highest degree term of the divisor is $x$ . Therefore, we divide $48x^3$ by $x$ to get $48x^2$ .

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 $x - 7$ by $48x^2$ to get $48x^3 - 336x^2$ .

Step 3: Subtract the Result from the Dividend

We then subtract the result obtained in the previous step from the dividend. In this case, we subtract $48x^3 - 336x^2$ from $48x^3 - 274x^2 - 427x - 41$ to get $-62x^2 - 427x - 41$ .

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 $-62x^2$ and the highest degree term of the divisor is $x$ . Therefore, we divide $-62x^2$ by $x$ to get $-62x$ .

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 $x - 7$ by $-62x$ to get $-62x^2 + 434x$ .

Step 6: Subtract the Result from the Dividend

We then subtract the result obtained in the previous step from the dividend. In this case, we subtract $-62x^2 + 434x$ from $-62x^2 - 427x - 41$ to get $-861x - 41$ .

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 $-861x$ and the highest degree term of the divisor is $x$ . Therefore, we divide $-861x$ by $x$ to get $-861$ .

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 $x - 7$ by $-861$ to get $-861x + 6037$ .

Step 9: Subtract the Result from the Dividend

We then subtract the result obtained in the previous step from the dividend. In this case, we subtract $-861x + 6037$ from $-861x - 41$ to get $-6078$ .

Conclusion

We have now completed the polynomial long division process. The quotient is $48x^2 - 62x - 861$ and the remainder is $-6078$ .

Final Answer

The final answer is $\boxed{48x^2 - 62x - 861, -6078}$ .

Discussion

Polynomial long division is a powerful tool in algebra that helps us to simplify complex expressions and find the quotient and remainder when a polynomial is divided by another polynomial. In this article, we used polynomial long division to find the quotient and remainder when the polynomial $48x^3 - 274x^2 - 427x - 41$ is divided by $x - 7$ . The quotient is $48x^2 - 62x - 861$ and the remainder is $-6078$ .

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Q: What is polynomial long division?

A: Polynomial long division is a method used to divide a polynomial by another polynomial of a lower degree. It is a powerful tool in algebra that helps us to simplify complex expressions and find the quotient and remainder when a polynomial is divided by another polynomial.

Q: What are the steps involved in polynomial long division?

A: The steps involved in polynomial long 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 the degree of the remainder is less than the degree of the divisor.

Q: What is the quotient in polynomial long division?

A: The quotient in polynomial long division is the result obtained by dividing the dividend by the divisor. It is the polynomial that results from the division process.

Q: What is the remainder in polynomial long division?

A: The remainder in polynomial long division is the result obtained after the division process is complete. It is the polynomial that is left over after the division process is complete.

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

A: You know when to stop the division process when the degree of the remainder is less than the degree of the divisor. At this point, the division process is complete and the quotient and remainder can be determined.

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

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

Not dividing the highest degree term of the dividend by the highest degree term of the divisor.
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 the degree of the remainder is less than the degree of the divisor.

Q: How do I check my work in polynomial long division?

A: To check your work in polynomial long division, you can multiply the quotient by the divisor and add the remainder to the product. If the result is equal to the original dividend, then your work is correct.

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

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

Simplifying complex expressions in algebra and calculus.
Finding the quotient and remainder when a polynomial is divided by another polynomial.
Solving systems of equations and inequalities.
Modeling real-world phenomena using polynomials and rational functions.

Q: How do I practice polynomial long division?

A: To practice polynomial long division, you can try the following:

Use online resources and practice problems to help you understand the process.
Work through examples and exercises in your textbook or online.
Practice dividing polynomials by hand or using a calculator.
Try dividing polynomials with different degrees and coefficients.

Q: What are some common mistakes to avoid when practicing polynomial long division?

A: Some common mistakes to avoid when practicing polynomial long division include:

Not following the steps of the process.
Not checking your work.
Not using the correct notation and terminology.
Not practicing regularly to build your skills and confidence.

Conclusion

Polynomial long division is a powerful tool in algebra that helps us to simplify complex expressions and find the quotient and remainder when a polynomial is divided by another polynomial. By understanding the steps involved in polynomial long division and practicing regularly, you can become proficient in this important skill.