Verified Answer :

  • The expression is rewritten as a product of fractions using the rule \(\frac{A/B}{C/D} = \frac{A \cdot D}{B \cdot C}\).
  • Each polynomial is factored: \(x^2+4x+3 = (x+3)(x+1)\), \(x^2+x = x(x+1)\), and \(2x^2-3x+1 = (2x-1)(x-1)\).
  • Common factors \((x+1)\) and \((2x-1)\) are canceled from the numerator and denominator.
  • The remaining expression simplifies to \(\frac{(x+3)(x-1)}{x}\), which is \(\boxed{\frac{x^2+2x-3}{x}}\).

Explanation

  1. Analysis of the Rational Expression

The given problem requires simplifying a complex rational expression, which is a division of two fractions:
\[\frac{\frac{x^2+4x+3}{2x-1}}{\frac{x^2+x}{2x^2-3x+1}}\]
The expression consists of four distinct polynomials. To simplify this, we first recall the rule for dividing fractions: \(\frac{A/B}{C/D} = \frac{A}{B} \cdot \frac{D}{C}\). Therefore, our expression can be rewritten as:
\[\frac{x^2+4x+3}{2x-1} \cdot \frac{2x^2-3x+1}{x^2+x}\]
where the denominators \(2x-1\), \(x^2+x\), and \(2x^2-3x+1\) must not be equal to zero.

  1. Factoring the Polynomials

Now, we proceed to factor each polynomial individually:

  1. For the first numerator: \(P_1 = x^2+4x+3\). We look for two numbers that multiply to 3 and add to 4, which are 3 and 1. Thus, \(P_1 = (x+3)(x+1)\).
  2. For the second denominator: \(P_4 = 2x^2-3x+1\). We look for factors of the form \((ax+b)(cx+d)\) such that \(ac=2\), \(bd=1\), and \(ad+bc=-3\). This leads us to \(P_4 = (2x-1)(x-1)\).
  3. For the third denominator: \(P_3 = x^2+x\). This is a simple case of factoring out the greatest common factor: \(P_3 = x(x+1)\).

Substituting these factored forms back into our expression:
\[\frac{(x+3)(x+1)}{2x-1} \cdot \frac{(2x-1)(x-1)}{x(x+1)}\]

  1. Simplifying and Final Result

With the expression factored, we can identify common factors in the numerator and the denominator:
\[\frac{(x+3)(x+1)}{(2x-1)} \cdot \frac{(2x-1)(x-1)}{x(x+1)}\]
We observe that \((x+1)\) appears in both the numerator and denominator, and \((2x-1)\) also appears in both. By cancelling these common factors (under the assumption that \(x eq -1\), \(x eq 1/2\), and \(x eq 0\)), the expression simplifies significantly:
\[\frac{(x+3) \cdot 1}{1} \cdot \frac{1 \cdot (x-1)}{x} = \frac{(x+3)(x-1)}{x}\]
Finally, we expand the numerator to get the simplified result:
\[\frac{x^2 - x + 3x - 3}{x} = \frac{x^2 + 2x - 3}{x}\]
This can also be expressed as \(x + 2 - \frac{3}{x}\). Both forms are acceptable as a simplified expression.

Examples

Rational expressions are frequently used in engineering and physics, for example, to model the efficiency of a system as a function of an input variable \(x\). When engineers derive formulas for systems, they often end up with complex fractional expressions. By simplifying these expressions, they can much more easily calculate performance values or identify critical points (like when the efficiency becomes zero or where it is undefined) without needing to perform redundant arithmetic. For instance, in an electrical circuit, impedance might be expressed as a ratio of polynomials, and simplifying it helps in determining the resonance frequency of the circuit.

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