(p^3-6)/(p-1)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (p^3 - 6) / (p - 1)

The expression (p^3 - 6) / (p - 1) represents a rational function, meaning it's a fraction where both the numerator and denominator are polynomials. To simplify this expression, we can use polynomial long division.

Polynomial Long Division

1. Set up the division: Write the numerator (p^3 - 6) as the dividend and the denominator (p - 1) as the divisor.

       _________
   p - 1 | p^3 - 6
   

2. Divide the leading terms: The leading term of the dividend (p^3) is divided by the leading term of the divisor (p). This gives us p^2.

       p^2 ______
   p - 1 | p^3 - 6
           p^3 - p^2
           -------
   

3. Subtract and bring down the next term: Subtract the result (p^3 - p^2) from the dividend. Bring down the next term (-6).

       p^2 ______
   p - 1 | p^3 - 6
           p^3 - p^2
           -------
               p^2 - 6
   

4. Repeat the process: Divide the leading term of the new dividend (p^2) by the leading term of the divisor (p). This gives us p.

       p^2 + p ______
   p - 1 | p^3 - 6
           p^3 - p^2
           -------
               p^2 - 6
               p^2 - p 
               -------
                   - p - 6 
   

5. Continue until the degree of the remainder is less than the degree of the divisor: Repeat the process, dividing -p by p, and bring down -6.

       p^2 + p - 1 ______
   p - 1 | p^3 - 6
           p^3 - p^2
           -------
               p^2 - 6
               p^2 - p 
               -------
                   - p - 6
                   - p + 1
                   -------
                       -7
   

Result

The quotient is p^2 + p - 1, and the remainder is -7. Therefore, we can express the original expression as:

(p^3 - 6) / (p - 1) = p^2 + p - 1 - 7/(p - 1)

This simplified form is equivalent to the original expression for all values of p except for p = 1 (where the denominator would be zero, making the expression undefined).