How to do long division method in polynomials?

Polynomial long division is a method for dividing one polynomial by another polynomial of equal or lower degree. Here’s a step-by-step guide on how to perform polynomial long division, incorporating insights from the provided reference which demonstrates the process (YouTube reference).

Steps for Polynomial Long Division

Set up the division: Write the dividend (the polynomial being divided) inside the long division symbol and the divisor (the polynomial doing the dividing) outside. Ensure both polynomials are written in descending order of exponents. If any terms are missing (e.g., no x term), include them with a coefficient of 0 as a placeholder.
Divide the first terms: Divide the first term of the dividend by the first term of the divisor. Write the result above the division symbol, aligned with the term in the dividend that has the same degree.
Multiply: Multiply the result from the previous step by the entire divisor. Write the product below the dividend, aligning like terms.
Subtract: Subtract the product from the corresponding terms in the dividend. Remember to change the signs of the terms being subtracted.
Bring down the next term: Bring down the next term of the dividend and write it next to the result of the subtraction.
Repeat: Repeat steps 2-5 until all terms of the dividend have been brought down and divided.
Determine the remainder: If, after bringing down the last term, the degree of the remaining polynomial is less than the degree of the divisor, then that polynomial is the remainder.

Example

Let's consider a simplified example inspired by the YouTube reference. Imagine we're dividing (x² - x + 1) by (x - 2).

Set up:
```
         _________
x - 2 | x^2 - x + 1
```
Divide the first terms: x² / x = x. Write 'x' above the -x term.
```
         x ______
x - 2 | x^2 - x + 1
```

Multiply: x * (x - 2) = x² - 2x. Write this below the dividend.

         x ______
x - 2 | x^2 - x + 1
       x^2 - 2x

Subtract: (x² - x) - (x² - 2x) = x. Remember to change signs and combine like terms.

         x ______
x - 2 | x^2 - x + 1
       x^2 - 2x
       -------
            x

Bring down the next term: Bring down the '+ 1'.

         x ______
x - 2 | x^2 - x + 1
       x^2 - 2x
       -------
            x + 1

Repeat: x / x = 1. Write '+ 1' next to the 'x' above.

         x + 1 ___
x - 2 | x^2 - x + 1
       x^2 - 2x
       -------
            x + 1

Multiply: 1 * (x - 2) = x - 2.

         x + 1 ___
x - 2 | x^2 - x + 1
       x^2 - 2x
       -------
            x + 1
            x - 2

Subtract: (x + 1) - (x - 2) = 3

         x + 1 ___
x - 2 | x^2 - x + 1
       x^2 - 2x
       -------
            x + 1
            x - 2
            -----
                3

Remainder: The remainder is 3.

Therefore, (x² - x + 1) / (x - 2) = x + 1 + (3 / (x - 2)).

Key Considerations

Placeholders: Always include placeholders (terms with a coefficient of 0) for any missing terms in the dividend. This maintains proper alignment during the subtraction steps.
Sign Changes: Be meticulous with sign changes when subtracting. This is a common area for errors.
Organization: Keeping the terms neatly aligned is crucial for avoiding mistakes.

According to the YouTube reference provided, the presenter performs polynomial long division by systematically dividing, multiplying, subtracting, and bringing down terms until the remainder is of a lower degree than the divisor. The presenter underscores the importance of meticulousness, particularly with the signs of numbers when calculating. For example, cancelling -x² requires multiplying correctly. Combining -8x² and -x² yields -9x².

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How to do long division method in polynomials?

Steps for Polynomial Long Division

Example

Key Considerations

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