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(x^3+x^2+3x-4)/(x^2+2x+1)

5 min read Jun 17, 2024
(x^3+x^2+3x-4)/(x^2+2x+1)

Analyzing the Rational Expression: (x³ + x² + 3x - 4) / (x² + 2x + 1)

This article will explore the rational expression (x³ + x² + 3x - 4) / (x² + 2x + 1) by performing polynomial long division to simplify the expression and identify its key features.

Polynomial Long Division

To simplify this expression, we can use polynomial long division. Here's how it works:

  1. Set up the division:

         __________
x² + 2x + 1 | x³ + x² + 3x - 4

  2. Divide the leading terms:

    • Divide x³ (the leading term of the dividend) by x² (the leading term of the divisor). This gives us 'x'.
    • Write 'x' above the x² term in the quotient.
         x        
x² + 2x + 1 | x³ + x² + 3x - 4

  3. Multiply the divisor by the quotient term:

    • Multiply (x² + 2x + 1) by 'x'. This gives us x³ + 2x² + x.
         x        
x² + 2x + 1 | x³ + x² + 3x - 4
             x³ + 2x² + x

  4. Subtract:

    • Subtract the result from the dividend.
         x        
x² + 2x + 1 | x³ + x² + 3x - 4
             x³ + 2x² + x
             ---------
                  -x² + 2x - 4

  5. Bring down the next term:

    • Bring down the '-4' from the dividend.
         x        
x² + 2x + 1 | x³ + x² + 3x - 4
             x³ + 2x² + x
             ---------
                  -x² + 2x - 4

  6. Repeat steps 2-5:

    • Divide the leading term of the new dividend (-x²) by the leading term of the divisor (x²). This gives us '-1'.
    • Multiply the divisor by '-1' and subtract.
         x - 1    
x² + 2x + 1 | x³ + x² + 3x - 4
             x³ + 2x² + x
             ---------
                  -x² + 2x - 4
                  -x² - 2x - 1
                  ---------
                          4x - 3

  7. Stop when the degree of the remainder is less than the degree of the divisor:

    • The degree of the remainder (4x - 3) is 1, which is less than the degree of the divisor (x² + 2x + 1, which is 2).

Simplified Expression

The result of the long division is:

(x³ + x² + 3x - 4) / (x² + 2x + 1) = x - 1 + (4x - 3) / (x² + 2x + 1)

Key Features

  • Simplified Form: The expression is now simplified into a quotient (x - 1) and a remainder term (4x - 3) / (x² + 2x + 1).
  • Asymptotes: The divisor, (x² + 2x + 1), can be factored as (x + 1)². This indicates a vertical asymptote at x = -1 (where the denominator becomes zero). The degree of the numerator (1) is less than the degree of the denominator (2), so there is a horizontal asymptote at y = 0.
  • Discontinuity: The rational expression has a hole at x = -1, because (x + 1) is a factor of both the numerator and denominator.
  • End Behavior: As x approaches positive or negative infinity, the expression approaches the horizontal asymptote y = 0.

By performing polynomial long division, we have gained a deeper understanding of the rational expression, identifying its simplified form, key features, and behavior.

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