To create a rational expression equivalent to [tex]\(\frac{5x^2 + 25x + 20}{7x}\)[/tex], we need to follow a step-by-step process that involves simplifying the numerator and the denominator, identifying common factors, and verifying the right parts of the expression.
1. Simplify the given numerator [tex]\(5x^2 + 25x + 20\)[/tex]:
First, we factorize the numerator.
[tex]\[
5x^2 + 25x + 20
\][/tex]
We look for factors of [tex]\(5x^2\)[/tex] and [tex]\(20\)[/tex] that add up to [tex]\(25x\)[/tex]. Rewriting it:
[tex]\[
5(x^2 + 5x + 4)
\][/tex]
Next, we factorize the quadratic expression [tex]\(x^2 + 5x + 4\)[/tex]:
[tex]\[
x^2 + 5x + 4 = (x + 4)(x + 1)
\][/tex]
Thus:
[tex]\[
5(x^2 + 5x + 4) = 5(x + 4)(x + 1)
\][/tex]
2. Write the simplified numerator:
[tex]\[
\text{Numerator: } 5(x + 4)(x + 1)
\][/tex]
3. Simplify and identify the denominator [tex]\(7x\)[/tex]:
The denominator remains [tex]\(7x\)[/tex]. This is already in its simplest form.
4. Write the given rational expression correctly factoring out pieces:
The expression can be written as:
[tex]\[
\frac{5(x + 4)(x + 1)}{7x}
\][/tex]
5. Identify the pieces from the provided choices:
- [tex]\(5 (x - 1)\)[/tex] - incorrect because it doesn't match our factors.
- [tex]\(x + 4\)[/tex] - correct, as it matches one of our factors.
- [tex]\(x - 1\)[/tex] - incorrect because it doesn't match our factors.
- [tex]\(5x^2 + 15x - 20\)[/tex] - incorrect, it's not in our numerator.
- [tex]\(7x\)[/tex] - correct, it matches our denominator.
- [tex]\(x^2 + 2x + 1\)[/tex] - incorrect, it doesn't match our numerator's factors.
- [tex]\(7x^2 + 7x\)[/tex] - incorrect, it's more complex than our denominator.
This leads us to conclude that the pieces for reconstructing the given rational expression are:
- In the numerator, use [tex]\(5(x + 4)\)[/tex] and [tex]\((x + 1)\)[/tex].
- In the denominator, use [tex]\(7x\)[/tex].
Therefore, placing the pieces correctly, we have:
[tex]\[
\frac{5(x + 4)(x + 1)}{7x}
\][/tex]
which matches [tex]\(\frac{5x^2 + 25x + 20}{7x}\)[/tex].
