(x^4-1)/(x-1)

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression: (x^4 - 1)/(x-1)

This expression involves a polynomial division, where the numerator is a fourth-degree polynomial and the denominator is a linear polynomial. We can simplify this expression using various methods.

Method 1: Polynomial Long Division

This method involves dividing the numerator by the denominator using long division, similar to the way we divide numbers. Here's how it works:

1. Set up the division:

        _______
   x-1 | x^4 + 0x^3 + 0x^2 + 0x - 1
   

2. Divide the leading terms: x^4 / x = x^3 Write x^3 above the line.

3. Multiply the quotient (x^3) by the divisor (x-1): x^3 * (x-1) = x^4 - x^3 Write this result below the dividend.

4. Subtract:

        _______
   x-1 | x^4 + 0x^3 + 0x^2 + 0x - 1
         x^4 - x^3
         -------
               x^3 + 0x^2
   

5. Bring down the next term:

        _______
   x-1 | x^4 + 0x^3 + 0x^2 + 0x - 1
         x^4 - x^3
         -------
               x^3 + 0x^2 + 0x
   

6. Repeat steps 2-5: x^3 / x = x^2 x^2 * (x-1) = x^3 - x^2

        _______
   x-1 | x^4 + 0x^3 + 0x^2 + 0x - 1
         x^4 - x^3
         -------
               x^3 + 0x^2 + 0x
               x^3 - x^2 
               -------
                     x^2 + 0x
   

7. Continue the process until the degree of the remainder is less than the degree of the divisor:

        _______
   x-1 | x^4 + 0x^3 + 0x^2 + 0x - 1
         x^4 - x^3
         -------
               x^3 + 0x^2 + 0x
               x^3 - x^2 
               -------
                     x^2 + 0x - 1
                     x^2 - x
                     -------
                           x - 1 
                           x - 1
                           -------
                               0 
   

8. The simplified expression is: (x^4 - 1)/(x-1) = x^3 + x^2 + x + 1

Method 2: Factoring

We can also simplify the expression by factoring both the numerator and the denominator.

1. Factor the numerator: x^4 - 1 is a difference of squares: x^4 - 1 = (x^2 + 1)(x^2 - 1) Furthermore, (x^2 - 1) is another difference of squares: (x^2 + 1)(x^2 - 1) = (x^2 + 1)(x + 1)(x - 1)

2. Cancel the common factor: (x^4 - 1)/(x-1) = [(x^2 + 1)(x + 1)(x - 1)]/(x-1) = (x^2 + 1)(x + 1)

Conclusion

Both methods lead to the same simplified expression. However, factoring is generally faster and more efficient for simpler expressions like this one. The simplified expression (x^2 + 1)(x + 1) is equivalent to x^3 + x^2 + x + 1, but it might be more convenient to leave it in factored form depending on the context.