How to do Long Division Method in Factorization?

The long division method, in the context of factorization, is primarily used to determine if a polynomial is a factor of another polynomial and, if so, to find the other factor. Here's how it works:

Key Idea: We use long division to divide a given polynomial (the dividend) by a known factor (the divisor). If the remainder is zero, then the divisor is indeed a factor, and the quotient we obtain is the other factor.

Steps involved, incorporating the reference information:

Arrange the Polynomials: First, arrange both the dividend and the divisor in descending order of their exponents. This means starting with the term having the highest power of the variable and proceeding towards the constant term.
- For example, if you have 3x + 2x^3 - 1 + x^2, rearrange it as 2x^3 + x^2 + 3x - 1.
Handle Missing Terms: Replace any missing terms (exponents) with zero. This maintains the proper place value during the division process.
- For instance, if you have x^4 - 1, rewrite it as x^4 + 0x^3 + 0x^2 + 0x - 1.
Set up the Long Division: Write the division problem similar to how you would with numbers. The polynomial to be factored (dividend) goes inside the division symbol, and the known factor (divisor) goes outside.
Divide the First Terms: Divide the first term of the dividend by the first term of the divisor. The result becomes the first term of the quotient (the answer).
Multiply: Multiply the entire divisor by the term you just wrote in the quotient.
Subtract: Subtract the result from the corresponding terms of the dividend.
Bring Down: Bring down the next term from the dividend.
Repeat: Repeat steps 4-7 until you've brought down all the terms from the dividend.
Check the Remainder:
- If the remainder is zero: The divisor is a factor of the dividend, and the quotient is the other factor. You've successfully factored the polynomial.
- If the remainder is not zero: The divisor is not a factor of the dividend.

Example:

Let's say we want to factor x^2 - 1 and we suspect that x - 1 is a factor.

Arrangement: Both are already arranged in descending order.
Missing Terms: x^2 - 1 can be written as x^2 + 0x - 1.
Setup:
```
       _________
x - 1 | x^2 + 0x - 1
```
Divide: x^2 / x = x
```
       x _______
x - 1 | x^2 + 0x - 1
```
Multiply: x * (x - 1) = x^2 - x

Subtract: (x^2 + 0x) - (x^2 - x) = x

       x _______
x - 1 | x^2 + 0x - 1
       -(x^2 - x)
       ---------
              x

Bring Down: Bring down the -1.

       x _______
x - 1 | x^2 + 0x - 1
       -(x^2 - x)
       ---------
              x - 1

Repeat: x / x = 1

       x + 1 ____
x - 1 | x^2 + 0x - 1
       -(x^2 - x)
       ---------
              x - 1

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

Subtract: (x - 1) - (x - 1) = 0

       x + 1 ____
x - 1 | x^2 + 0x - 1
       -(x^2 - x)
       ---------
              x - 1
              -(x - 1)
              -------
                  0

Remainder: The remainder is 0.

Conclusion:

Since the remainder is 0, x - 1 is a factor of x^2 - 1, and the other factor is x + 1. Therefore, x^2 - 1 = (x - 1)(x + 1).

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How to do Long Division Method in Factorization?

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