If 15 A 4 − 16 A 3 + 9 A 2 − 10 3 A + 6 15 A^4 - 16 A^3 + 9 A^2 - \frac{10}{3} A + 6 15 A 4 − 16 A 3 + 9 A 2 − 3 10 ​ A + 6 Is Divided By 3 A − 2 3 A - 2 3 A − 2 , Then The Quotient And Remainder Are, Respectively:(A) 5 A 3 − 2 A 2 + 5 A 5 A^3 - 2 A^2 + 5 A 5 A 3 − 2 A 2 + 5 A And 5 (B) 5 A 3 − 2 A 2 + 5 3 A 5 A^3 - 2 A^2 + \frac{5}{3} A 5 A 3 − 2 A 2 + 3 5 ​ A And 5 (C) $6

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, with numerous applications in various fields, including engineering, economics, and computer science. In this article, we will explore the process of dividing polynomials, with a focus on the given problem: $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$ divided by $3 a - 2$. We will determine the quotient and remainder, and discuss the significance of polynomial division in mathematics.

Understanding Polynomial Division

Polynomial division is a process of dividing a polynomial by another polynomial, resulting in a quotient and a remainder. 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. The product is then subtracted from the dividend, and the process is repeated until the degree of the remainder is less than the degree of the divisor.

The Given Problem

The given problem involves dividing the polynomial $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$ by $3 a - 2$. To solve this problem, we will use the process of polynomial division, which involves dividing the highest degree term of the dividend by the highest degree term of the divisor.

Step 1: Divide the Highest Degree Term

The highest degree term of the dividend is $15 a^4$, and the highest degree term of the divisor is $3 a$. To divide these two terms, we will divide $15 a^4$ by $3 a$, resulting in $5 a^3$.

Step 2: Multiply the Divisor by the Result

We will multiply the entire divisor $3 a - 2$ by the result $5 a^3$, resulting in $15 a^4 - 10 a^2$.

Step 3: Subtract the Product from the Dividend

We will subtract the product $15 a^4 - 10 a^2$ from the dividend $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$, resulting in $-16 a^3 + 19 a^2 - \frac{10}{3} a + 6$.

Step 4: Repeat the Process

We will repeat the process of dividing the highest degree term of the new dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result. We will continue this process until the degree of the remainder is less than the degree of the divisor.

Step 5: Determine the Quotient and Remainder

After repeating the process, we will determine the quotient and remainder. The quotient will be the result of the division, and the remainder will be the final result of the subtraction.

Solving the Problem

Let's apply the process of polynomial division to the given problem.

Step 1: Divide the Highest Degree Term

The highest degree term of the dividend is $15 a^4$, and the highest degree term of the divisor is $3 a$. To divide these two terms, we will divide $15 a^4$ by $3 a$, resulting in $5 a^3$.

Step 2: Multiply the Divisor by the Result

We will multiply the entire divisor $3 a - 2$ by the result $5 a^3$, resulting in $15 a^4 - 10 a^2$.

Step 3: Subtract the Product from the Dividend

We will subtract the product $15 a^4 - 10 a^2$ from the dividend $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$, resulting in $-16 a^3 + 19 a^2 - \frac{10}{3} a + 6$.

Step 4: Repeat the Process

We will repeat the process of dividing the highest degree term of the new dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result. We will continue this process until the degree of the remainder is less than the degree of the divisor.

Step 5: Determine the Quotient and Remainder

After repeating the process, we will determine the quotient and remainder. The quotient will be the result of the division, and the remainder will be the final result of the subtraction.

The Quotient and Remainder

After applying the process of polynomial division, we will determine the quotient and remainder.

The Quotient

The quotient will be $5 a^3 - 2 a^2 + \frac{5}{3} a$.

The Remainder

The remainder will be $5$.

Conclusion

In this article, we have explored the process of polynomial division, with a focus on the given problem: $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$ divided by $3 a - 2$. We have determined the quotient and remainder, and discussed the significance of polynomial division in mathematics. Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another, resulting in a quotient and a remainder. It is a crucial operation in mathematics, with numerous applications in various fields.

Final Answer

The final answer is:

• Quotient: $5 a^3 - 2 a^2 + \frac{5}{3} a$
• Remainder: $5$

Discussion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another, resulting in a quotient and a remainder. It is a crucial operation in mathematics, with numerous applications in various fields. In this article, we have explored the process of polynomial division, with a focus on the given problem: $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$ divided by $3 a - 2$. We have determined the quotient and remainder, and discussed the significance of polynomial division in mathematics.

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Polynomial Division" by Wolfram MathWorld
• [3] "Polynomial Division" by Khan Academy

Keywords

• Polynomial division
• Quotient
• Remainder
• Algebra
• Mathematics
  Polynomial Division Q&A ==========================

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another, resulting in a quotient and a remainder. In our previous article, we explored the process of polynomial division, with a focus on the given problem: $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$ divided by $3 a - 2$. We determined the quotient and remainder, and discussed the significance of polynomial division in mathematics. In this article, we will answer some frequently asked questions about polynomial division.

Q&A

Q: What is polynomial division?

A: Polynomial division is a process of dividing one polynomial by another, resulting in a quotient and a remainder.

Q: Why is polynomial division important?

A: Polynomial division is important because it allows us to simplify complex expressions and solve equations. It is a fundamental concept in algebra and has numerous applications in various fields, including engineering, economics, and computer science.

Q: How do I perform polynomial division?

A: To perform polynomial division, you need to divide the highest degree term of the dividend by the highest degree term of the divisor, and then multiply the entire divisor by the result. You will then subtract the product from the dividend, and repeat the process until the degree of the remainder is less than the degree of the divisor.

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

A: The quotient is the result of the division, and the remainder is the final result of the subtraction.

Q: Can I use polynomial division to solve equations?

A: Yes, polynomial division can be used to solve equations. By dividing both sides of the equation by a common factor, you can simplify the equation and solve for the unknown variable.

Q: Are there any rules or restrictions for polynomial division?

A: Yes, there are rules and restrictions for polynomial division. For example, the degree of the divisor must be less than the degree of the dividend, and the leading coefficient of the divisor must be non-zero.

Q: Can I use polynomial division to divide polynomials with complex coefficients?

A: Yes, polynomial division can be used to divide polynomials with complex coefficients. However, you need to be careful when working with complex numbers, as they can be difficult to handle.

Q: Are there any online tools or resources available for polynomial division?

A: Yes, there are many online tools and resources available for polynomial division, including calculators, software, and websites.

Examples

Example 1: Dividing $2x^3 + 3x^2 - 4x + 1$ by $x + 2$

To divide $2x^3 + 3x^2 - 4x + 1$ by $x + 2$, we need to divide the highest degree term of the dividend by the highest degree term of the divisor, and then multiply the entire divisor by the result. We will then subtract the product from the dividend, and repeat the process until the degree of the remainder is less than the degree of the divisor.

Example 2: Dividing $x^4 - 2x^3 + 3x^2 - 4x + 1$ by $x - 1$

To divide $x^4 - 2x^3 + 3x^2 - 4x + 1$ by $x - 1$, we need to divide the highest degree term of the dividend by the highest degree term of the divisor, and then multiply the entire divisor by the result. We will then subtract the product from the dividend, and repeat the process until the degree of the remainder is less than the degree of the divisor.

Conclusion

In this article, we have answered some frequently asked questions about polynomial division. Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another, resulting in a quotient and a remainder. It is a crucial operation in mathematics, with numerous applications in various fields. We have provided examples of polynomial division and discussed the rules and restrictions for polynomial division.

Final Answer

The final answer is:

• Quotient: $5 a^3 - 2 a^2 + \frac{5}{3} a$
• Remainder: $5$

Discussion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another, resulting in a quotient and a remainder. It is a crucial operation in mathematics, with numerous applications in various fields. In this article, we have explored the process of polynomial division, with a focus on the given problem: $15 a^4 - 16 a^3 + 9 a^2 - \frac{10}{3} a + 6$ divided by $3 a - 2$. We have determined the quotient and remainder, and discussed the significance of polynomial division in mathematics.

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Polynomial Division" by Wolfram MathWorld
• [3] "Polynomial Division" by Khan Academy

Keywords

• Polynomial division
• Quotient
• Remainder
• Algebra
• Mathematics