perform the indicated division (2x⁴+x³-19x²+18x+5)÷(2x-5)
pls write it in a simple way

To perform the division \((2x^4 + x^3 - 19x^2 + 18x + 5) \div (2x - 5)\), we can use **long division**.

### Steps:
1. Divide the first term of the dividend \(2x^4\) by the first term of the divisor \(2x\), which gives \(x^3\).
2. Multiply \(x^3\) by the divisor \((2x - 5)\), giving \(2x^4 - 5x^3\).
3. Subtract this result from the original polynomial:
   \[
   (2x^4 + x^3 - 19x^2 + 18x + 5) - (2x^4 - 5x^3) = 6x^3 - 19x^2 + 18x + 5
   \]
4. Divide \(6x^3\) by \(2x\), which gives \(3x^2\).
5. Multiply \(3x^2\) by the divisor \((2x - 5)\), giving \(6x^3 - 15x^2\).
6. Subtract:
   \[
   (6x^3 - 19x^2 + 18x + 5) - (6x^3 - 15x^2) = -4x^2 + 18x + 5
   \]
7. Divide \(-4x^2\) by \(2x\), which gives \(-2x\).
8. Multiply \(-2x\) by the divisor \((2x - 5)\), giving \(-4x^2 + 10x\).
9. Subtract:
   \[
   (-4x^2 + 18x + 5) - (-4x^2 + 10x) = 8x + 5
   \]
10. Divide \(8x\) by \(2x\), which gives \(4\).
11. Multiply \(4\) by the divisor \((2x - 5)\), giving \(8x - 20\).
12. Subtract:
    \[
    (8x + 5) - (8x - 20) = 25
    \]

### Final Result:
\[
(2x^4 + x^3 - 19x^2 + 18x + 5) \div (2x - 5) = x^3 + 3x^2 - 2x + 4 + \frac{25}{2x - 5}
\]

The quotient is \(x^3 + 3x^2 - 2x + 4\), with a remainder of \(25\), so the division can be expressed as:
\[
x^3 + 3x^2 - 2x + 4 + \frac{25}{2x - 5}
\]

Do you want more details or explanations on any step?

Here are 5 follow-up questions you might find interesting:
1. How does synthetic division compare with long division for polynomial division?
2. How can you check the result of a polynomial division?
3. What are the possible applications of polynomial division in calculus?
4. How would the remainder change if the divisor were \(x - 3\)?
5. What is the relationship between the quotient and the remainder in polynomial division?

**Tip:** Always check your division by multiplying the quotient and the divisor, and adding the remainder to see if you get back the original polynomial.

perform the indicated division (2x⁴+x³-19x²+18x+5)÷(2x-5) pls write it in a simple way

Learn how to perform the polynomial long division of (2x⁴ + x³ - 19x² + 18x + 5) divided by (2x - 5) with a step-by-step solution. This problem involves key concepts in polynomial division and algebra. Suitable for high school students in Grades 10-12.

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