7) Divide \($ \left(r^4-r^3-8r^2+r+12\right) \div (r+1) \) $.A) \[$ R^3-2r^2-6r+7+\frac{5}{r+1} \$\] $ B) \($ R^3-2r^2-6r+6+\frac{9}{r+1} \) $ C) \[$ R^3-2r^2-6r+6+\frac{10}{r+1} \$\] $ D)

by ADMIN 194 views

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is an essential tool for simplifying complex expressions and solving equations. In this article, we will focus on dividing a polynomial of degree 4 by a linear polynomial of degree 1. We will use the given polynomial (r4βˆ’r3βˆ’8r2+r+12)\left(r^4-r^3-8r^2+r+12\right) and divide it by (r+1)(r+1).

Step 1: Set Up the Division

To divide the polynomial (r4βˆ’r3βˆ’8r2+r+12)\left(r^4-r^3-8r^2+r+12\right) by (r+1)(r+1), we need to set up the division in the following format:

(r4βˆ’r3βˆ’8r2+r+12)(r+1)\frac{\left(r^4-r^3-8r^2+r+12\right)}{(r+1)}

Step 2: Divide the Leading Term

The leading term of the dividend is r4r^4, and the leading term of the divisor is rr. To divide the leading term of the dividend by the leading term of the divisor, we get:

r4r=r3\frac{r^4}{r} = r^3

Step 3: Multiply the Divisor by the Result

We multiply the divisor (r+1)(r+1) by the result r3r^3 to get:

(r+1)β‹…r3=r4+r3(r+1) \cdot r^3 = r^4 + r^3

Step 4: Subtract the Product from the Dividend

We subtract the product r4+r3r^4 + r^3 from the dividend (r4βˆ’r3βˆ’8r2+r+12)\left(r^4-r^3-8r^2+r+12\right) to get:

(r4βˆ’r3βˆ’8r2+r+12)βˆ’(r4+r3)=βˆ’2r3βˆ’8r2+r+12\left(r^4-r^3-8r^2+r+12\right) - (r^4 + r^3) = -2r^3 - 8r^2 + r + 12

Step 5: Repeat the Process

We repeat the process by dividing the leading term of the new dividend βˆ’2r3-2r^3 by the leading term of the divisor rr, which gives us:

βˆ’2r3r=βˆ’2r2\frac{-2r^3}{r} = -2r^2

Step 6: Multiply the Divisor by the Result

We multiply the divisor (r+1)(r+1) by the result βˆ’2r2-2r^2 to get:

(r+1)β‹…βˆ’2r2=βˆ’2r3βˆ’2r2(r+1) \cdot -2r^2 = -2r^3 - 2r^2

Step 7: Subtract the Product from the Dividend

We subtract the product βˆ’2r3βˆ’2r2-2r^3 - 2r^2 from the new dividend βˆ’2r3βˆ’8r2+r+12-2r^3 - 8r^2 + r + 12 to get:

βˆ’2r3βˆ’8r2+r+12βˆ’(βˆ’2r3βˆ’2r2)=βˆ’6r2+r+12-2r^3 - 8r^2 + r + 12 - (-2r^3 - 2r^2) = -6r^2 + r + 12

Step 8: Repeat the Process

We repeat the process by dividing the leading term of the new dividend βˆ’6r2-6r^2 by the leading term of the divisor rr, which gives us:

βˆ’6r2r=βˆ’6r\frac{-6r^2}{r} = -6r

Step 9: Multiply the Divisor by the Result

We multiply the divisor (r+1)(r+1) by the result βˆ’6r-6r to get:

(r+1)β‹…βˆ’6r=βˆ’6r2βˆ’6r(r+1) \cdot -6r = -6r^2 - 6r

Step 10: Subtract the Product from the Dividend

We subtract the product βˆ’6r2βˆ’6r-6r^2 - 6r from the new dividend βˆ’6r2+r+12-6r^2 + r + 12 to get:

βˆ’6r2+r+12βˆ’(βˆ’6r2βˆ’6r)=7r+12-6r^2 + r + 12 - (-6r^2 - 6r) = 7r + 12

Step 11: Repeat the Process

We repeat the process by dividing the leading term of the new dividend 7r7r by the leading term of the divisor rr, which gives us:

7rr=7\frac{7r}{r} = 7

Step 12: Multiply the Divisor by the Result

We multiply the divisor (r+1)(r+1) by the result 77 to get:

(r+1)β‹…7=7r+7(r+1) \cdot 7 = 7r + 7

Step 13: Subtract the Product from the Dividend

We subtract the product 7r+77r + 7 from the new dividend 7r+127r + 12 to get:

7r+12βˆ’(7r+7)=57r + 12 - (7r + 7) = 5

Conclusion

We have successfully divided the polynomial (r4βˆ’r3βˆ’8r2+r+12)\left(r^4-r^3-8r^2+r+12\right) by (r+1)(r+1) to get the result:

r3βˆ’2r2βˆ’6r+7+5r+1r^3-2r^2-6r+7+\frac{5}{r+1}

This result is in the form of a polynomial plus a fraction, which is a common outcome of polynomial division.

Answer

The correct answer is:

\boxed{A) r^3-2r^2-6r+7+\frac{5}{r+1}}$<br/> **Dividing Polynomials: A Q&A Guide** ===================================== **Q: What is polynomial division?** ------------------------------ A: Polynomial division is a mathematical operation that involves dividing one polynomial by another. It is a fundamental concept 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, solve equations, and find the roots of a polynomial. Polynomial division is a crucial tool in algebra that helps us break down complex expressions into simpler ones. **Q: What are the steps involved in polynomial division?** ------------------------------------------------ A: The steps involved in polynomial division are: 1. Set up the division in the correct format. 2. Divide the leading term of the dividend by the leading term of the divisor. 3. Multiply the divisor by the result and subtract the product from the dividend. 4. Repeat the process until the degree of the remainder is less than the degree of the divisor. **Q: What is the remainder in polynomial division?** ---------------------------------------------- A: The remainder in polynomial division is the expression that is left after the division process is complete. It is the part of the dividend that cannot be divided by the divisor. **Q: How do we handle fractions in polynomial division?** --------------------------------------------------- A: When we encounter a fraction in polynomial division, we can multiply the numerator and denominator by the same value to eliminate the fraction. This is known as "clearing the fraction." **Q: What is the difference between polynomial division and long division?** ---------------------------------------------------------------- A: Polynomial division and long division are both methods of dividing numbers, but they are used in different contexts. Polynomial division is used to divide polynomials, while long division is used to divide integers. **Q: Can we divide a polynomial by a polynomial of a higher degree?** ---------------------------------------------------------------- A: No, we cannot divide a polynomial by a polynomial of a higher degree. This is because the degree of the remainder would be higher than the degree of the divisor, which is not allowed in polynomial division. **Q: How do we check the result of polynomial division?** --------------------------------------------------- A: To check the result of polynomial division, we can multiply the divisor by the quotient and add the remainder. If the result is equal to the dividend, then the division is correct. **Q: What are some common mistakes to avoid in polynomial division?** ---------------------------------------------------------------- A: Some common mistakes to avoid in polynomial division include: * Not setting up the division in the correct format. * Not multiplying the divisor by the result correctly. * Not subtracting the product from the dividend correctly. * Not handling fractions correctly. **Q: How do we use polynomial division in real-world applications?** ---------------------------------------------------------------- A: Polynomial division is used in a variety of real-world applications, including: * Engineering: Polynomial division is used to design and analyze complex systems, such as electrical circuits and mechanical systems. * Computer Science: Polynomial division is used in computer algorithms, such as the Euclidean algorithm, to find the greatest common divisor of two numbers. * Economics: Polynomial division is used to model and analyze economic systems, such as supply and demand curves. **Conclusion** ---------- Polynomial division is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. By understanding the steps involved in polynomial division, we can apply it to a variety of real-world applications.