Divide. 8 X 5 + 3 X 4 + 12 2 X 2 \frac{8x^5 + 3x^4 + 12}{2x^2} 2 X 2 8 X 5 + 3 X 4 + 12 ​

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Introduction

Algebraic division is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we will focus on dividing a polynomial expression by another polynomial expression. Specifically, we will divide the expression $\frac{8x^5 + 3x^4 + 12}{2x^2}$ using the long division method.

Understanding the Problem

To divide the given expression, we need to understand the concept of polynomial division. Polynomial division is a process of dividing a polynomial expression by another polynomial expression. 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.

Step 1: Divide the Highest Degree Term

The first step in dividing the expression $\frac{8x^5 + 3x^4 + 12}{2x^2}$ is to divide the highest degree term of the dividend, which is $8x^5$, by the highest degree term of the divisor, which is $2x^2$. This will give us the first term of the quotient.

Step 3: Multiply the Divisor by the Result

Once we have the first term of the quotient, we need to multiply the entire divisor by the result. In this case, we will multiply $2x^2$ by $4x^3$, which gives us $8x^5$.

Step 4: Subtract the Result from the Dividend

Next, we need to subtract the result from the dividend. In this case, we will subtract $8x^5$ from $8x^5 + 3x^4 + 12$, which gives us $3x^4 + 12$.

Step 5: Repeat the Process

We will repeat the process of dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the entire divisor by the result, and subtracting the result from the dividend until we have a remainder that is less than the degree of the divisor.

Step 6: Divide the Next Term

The next term in the dividend is $3x^4$. We will divide $3x^4$ by $2x^2$, which gives us $\frac{3x^4}{2x^2} = \frac{3}{2}x^2$.

Step 7: Multiply the Divisor by the Result

We will multiply the entire divisor by the result, which gives us $2x^2 \times \frac{3}{2}x^2 = 3x^4$.

Step 8: Subtract the Result from the Dividend

We will subtract the result from the dividend, which gives us $3x^4 + 12 - 3x^4 = 12$.

Step 9: Divide the Remainder

Since the remainder is $12$, which is less than the degree of the divisor, we will stop the division process.

Conclusion

In conclusion, we have successfully divided the expression $\frac{8x^5 + 3x^4 + 12}{2x^2}$ using the long division method. The quotient is $4x^3 + \frac{3}{2}x^2$ and the remainder is $12$.

Final Answer

The final answer is $\boxed{4x^3 + \frac{3}{2}x^2}$.

Related Topics

• Polynomial division
• Long division method
• Algebraic division

References

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

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Introduction

In our previous article, we discussed how to divide the expression $\frac{8x^5 + 3x^4 + 12}{2x^2}$ using the long division method. In this article, we will answer some frequently asked questions related to polynomial division and provide additional examples to help you understand the concept better.

Q&A

Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial expression by another polynomial expression. It 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.

Q: What is the long division method?

A: The long division method is a step-by-step process of dividing a polynomial expression by another polynomial expression. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the entire divisor by the result, and subtracting the result from the dividend.

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

A: You should stop the division process when the remainder is less than the degree of the divisor. This is because the remainder will be a polynomial expression of a lower degree than the divisor.

Q: Can I use polynomial division to divide a polynomial expression by a constant?

A: Yes, you can use polynomial division to divide a polynomial expression by a constant. In this case, the constant will be the divisor, and the polynomial expression will be the dividend.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the result of the division process when the remainder is less than the degree of the divisor. It is a polynomial expression of a lower degree than the divisor.

Q: Can I use polynomial division to divide a polynomial expression by another polynomial expression with a higher degree?

A: No, you cannot use polynomial division to divide a polynomial expression by another polynomial expression with a higher degree. In this case, the division process will not terminate, and you will not be able to find a quotient.

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

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

Examples

Example 1: Divide $x^3 + 2x^2 + 3x + 4$ by $x + 2$

To divide $x^3 + 2x^2 + 3x + 4$ by $x + 2$, we will use the long division method.

x + 2 | x^3 + 2x^2 + 3x + 4
  - (x^3 + 2x^2)
  -------------------
  -x^2 + 3x + 4
  - (-x^2 - 2x)
  -------------------
  x + 4
  - (x + 2)
  -------------------
  2

The quotient is $x^2 + x - 1$ and the remainder is $2$.

Example 2: Divide $x^4 + 2x^3 + 3x^2 + 4x + 5$ by $x^2 + 2x + 1$

To divide $x^4 + 2x^3 + 3x^2 + 4x + 5$ by $x^2 + 2x + 1$, we will use the long division method.

x^2 + 2x + 1 | x^4 + 2x^3 + 3x^2 + 4x + 5
  - (x^4 + 2x^3 + x^2)
  -------------------
  x^2 + 4x + 5
  - (x^2 + 2x + 1)
  -------------------
  2x + 4
  - (2x + 2)
  -------------------
  2

The quotient is $x^2 + 2x + 1$ and the remainder is $2$.

Conclusion

In conclusion, polynomial division is a powerful tool for dividing polynomial expressions. By using the long division method, you can divide a polynomial expression by another polynomial expression and find the quotient and remainder. Remember to check your work by multiplying the quotient by the divisor and adding the remainder.

Final Answer

The final answer is $\boxed{4x^3 + \frac{3}{2}x^2}$.

Related Topics

• Polynomial division
• Long division method
• Algebraic division

References

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