Answered

Simplify the expression:



$\frac{\frac{4 x^2+2 x}{x^2+x-2}}{\frac{8 x^2+4 x}{3 x^2+10 x+8}}$


A$\frac{3 x+4}{2(x-1)}$


B$\frac{3 x+4}{2(x+1)}$


C$\frac{3 x-4}{2(x-1)}$


D$3 x+4$

Verified Answer :

  • Factor each polynomial within the expression to its simplest form.
  • Rewrite the division as multiplication by the reciprocal.
  • Cancel common factors across the numerator and denominator.
  • The simplified form is \(\boxed{\frac{3x+4}{2(x-1)}}\).

Explanation

  1. Factoring the Polynomials

We are asked to simplify the rational expression:
\[ \frac{\frac{4x^2+2x}{x^2+x-2}}{\frac{8x^2+4x}{3x^2+10x+8}} \]
First, we analyze each component polynomial by factoring it.

  • For \(4x^2+2x\), we factor out the common term \(2x\):

\[ 4x^2+2x = 2x(2x+1) \]

  • For \(x^2+x-2\), we look for two numbers that multiply to \(-2\) and add to \(1\), which are \(2\) and \(-1\):

\[ x^2+x-2 = (x+2)(x-1) \]

  • For \(8x^2+4x\), we factor out the common term \(4x\):

\[ 8x^2+4x = 4x(2x+1) \]

  • For \(3x^2+10x+8\), we look for factors of the form \((ax+b)(cx+d)\). By testing factors, we find:

\[ 3x^2+10x+8 = (3x+4)(x+2) \]
The expression is defined provided the denominators are non-zero.

  1. Simplifying the Expression

Now we rewrite the original division of fractions as multiplication by the reciprocal of the second fraction:
\[ \frac{2x(2x+1)}{(x+2)(x-1)} \cdot \frac{(3x+4)(x+2)}{4x(2x+1)} \]
Next, we simplify the expression by canceling the common factors found in both the numerators and denominators:

  • The term \((2x+1)\) is in both the numerator and denominator.
  • The term \((x+2)\) is in both the numerator and denominator.
  • The term \(x\) is in both the numerator and denominator.

Canceling these leaves us with:
\[ \frac{2}{x-1} \cdot \frac{3x+4}{4} \]
\[ = \frac{2(3x+4)}{4(x-1)} \]
\[ = \frac{3x+4}{2(x-1)} \]
This matches the first option provided.

  1. Final Result

After factoring the polynomials and simplifying the rational expression by canceling common factors, we determined that the expression reduces to \(\frac{3x+4}{2(x-1)}\). Comparing this to the given options confirms that this is the correct result.
Thus, the final answer is \(\boxed{\frac{3x+4}{2(x-1)}}\).

Examples

Rational expressions are commonly used in fields like physics and engineering to relate variables, such as calculating velocity or acceleration in systems involving complex ratios. For instance, if an engineer is designing a control system where a performance metric is given by a ratio of two complex polynomials, simplifying the expression helps in determining how a specific parameter (represented by \(x\)) influences the overall system output. By simplifying complex rational expressions, we can easily identify the behavior of the system and make necessary adjustments for efficiency and stability.

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