Answered

Drag each expression to the correct location on the model. Not all expressions will be used.



$\frac{5 x^2+25 x+20}{7 x}$



Determine where each piece below belongs to create a rational expression equivalent to the one shown above.



$x+4 \quad 7 x \quad 5(x-1) \quad x-1 \quad 5 x^2+15 x-20$

Verified Answer :

  • Factor the numerator: \(5x^2 + 25x + 20 = 5(x+1)(x+4)\).
  • Set up the rational expression as \(\frac{5(x+1)(x+4)}{7x}\).
  • Identify matching components: \(x+4\) belongs in the numerator and \(7x\) belongs in the denominator.
  • Conclude that \(\boxed{\frac{5(x+1)(x+4)}{7x}}\) is the equivalent form.

Explanation

  1. Factoring the Expression

We are given the rational expression \(\frac{5x^2 + 25x + 20}{7x}\) and asked to identify components to build an equivalent form. First, we factor the numerator:
\[5x^2 + 25x + 20 = 5(x^2 + 5x + 4)\]
Recognizing that the quadratic \(x^2 + 5x + 4\) factors as \((x+1)(x+4)\), we can write the entire numerator as:
\[5(x+1)(x+4)\]
Thus, the expression becomes:
\[\frac{5(x+1)(x+4)}{7x}\]

  1. Matching Components to the Model

Now we compare our derived components with the provided options: \(x+4\), \(7x\), \(5(x-1)\), \(x-1\), and \(5x^2+15x-20\).

  • The factor \((x+4)\) appears in the numerator of our factored form.
  • The term \(7x\) matches the denominator of the original fraction.
  • The terms \(5(x-1)\), \(x-1\), and \(5x^2+15x-20\) do not appear as factors or components of the simplified form of the given expression.

Therefore, the equivalent model is constructed using the numerator factors 5, \((x+1)\), and \((x+4)\) over the denominator \(7x\). Since only specific items were requested, we match the ones provided that align with this structure.

  1. Final Conclusion

Based on the factoring, the expression \(\frac{5x^2 + 25x + 20}{7x}\) is equivalent to:
\[\frac{5(x+1)(x+4)}{7x}\]
The expressions from the list that fit this model are \(x+4\) in the numerator and \(7x\) in the denominator. The others are not part of this specific factorization.

Examples

Rational expressions are frequently used in engineering and physics to describe relationships where one quantity depends on another, such as in impedance in electrical circuits or rate of change in fluid dynamics. Simplifying these expressions allows for easier calculations, just as factoring polynomials helps isolate key variables, making complex systems much easier to analyze and solve in practice.

Related Questions