To create a rational expression equivalent to [tex]\(\frac{5x^2 + 25x + 20}{7x}\)[/tex], we need to simplify the given expression by factoring both the numerator and the denominator where possible.
Let's break this down step-by-step:
1. Numerator Simplification:
The numerator is [tex]\(5x^2 + 25x + 20\)[/tex]. We first look for common factors and then factor the quadratic expression.
- Take out the common factor of 5:
[tex]\[
5x^2 + 25x + 20 = 5(x^2 + 5x + 4)
\][/tex]
- Factor the quadratic expression [tex]\(x^2 + 5x + 4\)[/tex]. We need to find two numbers that multiply to 4 (constant term) and add up to 5 (linear coefficient).
The two numbers that satisfy these conditions are 4 and 1.
[tex]\[
x^2 + 5x + 4 = (x + 4)(x + 1)
\][/tex]
- Thus, the factored form of the numerator is:
[tex]\[
5(x + 4)(x + 1)
\][/tex]
2. Denominator Simplification:
The denominator is [tex]\(7x\)[/tex], which is already in its simplest form.
So, the original expression [tex]\(\frac{5x^2 + 25x + 20}{7x}\)[/tex] can be expressed as:
[tex]\[
\frac{5(x+4)(x+1)}{7x}
\][/tex]
To match with the given pieces, we identify:
- Numerator terms: [tex]\(5(x + 4)(x + 1)\)[/tex]
- Denominator: [tex]\(7x\)[/tex]
Now, we map these to the correct pieces:
- [tex]\(5(x+4)\)[/tex] -> numerator factor 1
- [tex]\(x+1\)[/tex] -> numerator factor 2
- [tex]\(7x\)[/tex] -> denominator
So, the rational expression equivalent to the given one is:
[tex]\[
\frac{5(x+4)(x+1)}{7x}
\][/tex]
The correct components from the list are:
- [tex]\(x+4\)[/tex]
- [tex]\(7x\)[/tex]
We see that not all provided expressions are used, such as [tex]\(5(x-1)\)[/tex], [tex]\(x-1\)[/tex], and [tex]\(\frac{x^2 + 2x + 1}{\cdots}\)[/tex], because they do not fit into the simplified form of the original expression.
