To determine where each piece belongs to create a rational expression equivalent to
[tex]$\frac{5x^2 + 25x + 20}{7x}$[/tex]
we need to factorize the numerator and then match the components accordingly.
1. Factorize the numerator:
Given numerator: [tex]\(5x^2 + 25x + 20\)[/tex]
Notice that we can factor out a common factor of 5:
[tex]$5(x^2 + 5x + 4)$[/tex]
2. Factor the quadratic expression inside the parentheses:
We need to factorize [tex]\(x^2 + 5x + 4\)[/tex]:
The quadratic expression [tex]\(x^2 + 5x + 4\)[/tex] can be factored into:
[tex]$(x + 1)(x + 4)$[/tex]
3. Substitute the factored form back into the numerator:
So, the numerator becomes:
[tex]$5(x + 1)(x + 4)$[/tex]
4. Form the complete rational expression:
Combining the factored numerator with the denominator [tex]\(7x\)[/tex], we get:
[tex]$\frac{5(x + 1)(x + 4)}{7x}$[/tex]
5. Match each given piece to our rational expression:
- The numerator is [tex]\(5(x + 1)(x + 4)\)[/tex]
- The denominator is [tex]\(7x\)[/tex]
So, the needed pieces to construct the rational expression:
[tex]$\frac{5(x + 1)(x + 4)}{7x}$[/tex]
are:
- [tex]\(5(x + 1)(x + 4)\)[/tex]
- [tex]\(7x\)[/tex]
The other expressions provided ([tex]\(x - 1\)[/tex], [tex]\(5(x - 1)\)[/tex], [tex]\(5x^2 + 15x - 20\)[/tex], [tex]\(\frac{x^2 + 2x + 1}{7x^2 + 7x}\)[/tex]) are not needed to form the simplified version of the given rational expression.
Thus, the needed pieces are:
- [tex]\(5(x + 1)(x + 4)\)[/tex]
- [tex]\(7x\)[/tex]
