RÁPIDO e FÁCIL | BRIOT RUFFINI | divisão de polinômios

Summary

In this tutorial, Sandro explains the process of polynomial division using the Ruffini method, also known as the Fini Bobbin device. The video walks through the step-by-step division of a polynomial by a first-degree binomial, demonstrating how to identify coefficients, set up the table, and find both the quotient and remainder. Sandro also introduces the Remainder Theorem, showing how it can be used to quickly determine the remainder by substituting the root of the divisor into the polynomial. The video provides clear, practical examples and encourages viewers to practice and share their learning.

Takeaways

• 😀 The video explains the process of polynomial division using the synthetic division method, also known as the 'fini bobbin' device.
• 😀 Sandro demonstrates how to divide a polynomial by a first-degree binomial, such as x - 2, using synthetic division.
• 😀 The first step in synthetic division is to identify the structure of the divisor and confirm that it is a first-degree polynomial.
• 😀 Coefficients of the dividend are placed in a horizontal line, with missing terms completed by adding zeroes if necessary.
• 😀 The root of the divisor (x - 2) is found by setting the equation x - 2 = 0, which gives x = 2.
• 😀 Synthetic division involves iterative steps where the coefficients are processed to find the quotient and remainder.
• 😀 The first value is carried down and used for the subsequent calculations. Each step involves multiplying and adding the results.
• 😀 The process is repeated, checking for the final remainder after processing all the terms of the dividend.
• 😀 The remainder is identified as the last value obtained in the division process, and the quotient is formed from the other results.
• 😀 Sandro also mentions the Remainder Theorem, which can be used to find the remainder of any division by a first-degree polynomial (like x - 2) by substituting the root into the polynomial.
• 😀 The example using the Remainder Theorem confirms that substituting x = 2 into the polynomial gives a remainder of 4, verifying the division result.

Q & A

What is synthetic division used for in the video?

Synthetic division is used to divide a polynomial by a first-degree binomial, providing both the quotient and remainder of the division.

What is the first step in performing synthetic division?

The first step is to check the structure of the divisor, confirming that it is a first-degree binomial (like x - 2).

How do you set up the synthetic division table?

To set up the synthetic division table, draw a small vertical bar and a larger horizontal bar. Then, write down the coefficients of the dividend in the horizontal row, starting with the largest degree term.

What do you do if a term is missing in the polynomial (e.g., a missing x² term)?

If a term is missing, you fill in the corresponding coefficient with zero. For example, if there's no x² term, you place 0 in its position.

What is the role of the root of the divisor in synthetic division?

The root of the divisor is used to begin the synthetic division process. For the divisor x - 2, the root is 2, and it is placed at the start of the synthetic division process.

What is the process for performing the division after setting up the synthetic division table?

To perform the division, take the first coefficient of the dividend and multiply it by the root of the divisor. Then, add the result to the next coefficient, repeating this process for all terms.

How do you interpret the results of the synthetic division process?

The last number in the synthetic division table is the remainder, and the numbers in the row (excluding the remainder) represent the coefficients of the quotient polynomial.

What does the remainder theorem state, and how is it used?

The remainder theorem states that if a polynomial is divided by a first-degree binomial, the remainder is the value of the polynomial when the root of the divisor is substituted into the polynomial.

Can synthetic division be used for higher-degree polynomials? If so, how?

Yes, synthetic division can be used for higher-degree polynomials as long as the divisor is a first-degree binomial. The division process remains the same regardless of the degree of the polynomial.

How can you verify the remainder using synthetic division and the remainder theorem?

To verify the remainder, you can substitute the root of the divisor into the polynomial (as per the remainder theorem) and compare the result with the remainder obtained through synthetic division.