Division of Polynomials (Long Division of Polynomials)

Summary

The video demonstrates how to solve a polynomial division problem using long division. It walks through dividing the polynomial 2x³ + 3x² - 5x + 6 by x - 2. The process includes dividing the leading terms, multiplying, subtracting, and repeating until reaching the remainder. The final quotient is 2x² + 7x + 9, with a remainder of 24, which is expressed as a fraction over the divisor. The solution is presented as 2x² + 7x + 9 + 24/(x-2).

Takeaways

• 📝 The problem involves solving polynomial division using the long division method.
• 🔢 The dividend is 2x³ + 3x² - 5x + 6, and the divisor is x - 2.
• ➗ The first step is dividing 2x³ by x, resulting in 2x².
• 🔄 Multiplying 2x² by (x - 2) gives 2x³ - 4x², and subtracting it from the dividend results in 7x².
• 🔢 The next step is dividing 7x² by x, which gives 7x.
• 🔄 Multiplying 7x by (x - 2) gives 7x² - 14x, and subtracting it from the current terms results in 9x.
• ➕ Dividing 9x by x results in 9, which is then multiplied by (x - 2), yielding 9x - 18.
• 🔄 Subtracting 9x - 18 from the current terms results in a remainder of 24.
• 🧮 The quotient from the division is 2x² + 7x + 9, and the remainder is 24.
• ✅ The final result is expressed as 2x² + 7x + 9 + 24/(x - 2).

Q & A

What is the given polynomial division problem in the script?

The problem is to divide the polynomial 2x³ + 3x² - 5x + 6 by the divisor x - 2 using long division.

What is the first step in polynomial long division according to the script?

The first step is to divide the leading term of the dividend (2x³) by the leading term of the divisor (x), which results in 2x².

How do you multiply the first quotient term by the divisor?

You multiply 2x² by x - 2, which gives 2x³ - 4x².

What happens after subtracting the first multiplication result?

After subtracting 2x³ - 4x² from the original terms, you get 0 for the x³ term and 7x² after simplifying the second term.

What is done after bringing down the next term?

After bringing down -5x, the next step is to divide the new leading term (7x²) by the leading term of the divisor (x), which gives 7x.

What is the result after multiplying the second quotient term by the divisor?

Multiplying 7x by x - 2 results in 7x² - 14x.

How is the subtraction carried out in the second step of division?

After subtracting 7x² - 14x from the previous terms, you get 9x by simplifying -5x + 14x.

What is the quotient after the third division?

Dividing 9x by x results in the next quotient term, which is 9.

How is the final remainder calculated?

Multiplying 9 by x - 2 results in 9x - 18. After subtracting, the remainder becomes 24.

What is the final answer in polynomial long division?

The final answer is 2x² + 7x + 9 with a remainder of 24, which can be written as 2x² + 7x + 9 + 24/(x - 2).