Question

Consider the rational expression 2x+3/x^2+2x+3 . Which statements are true about the rational expression? Select each correct answer.

The numerator has two terms.
The denominator has three factors.
The denominator has three terms.
2x + 3 is a single factor in the numerator.
2x + 3 in the numerator divides out with 2x + 3 in the denominator.

Answer 1

The numerator has two terms.

The denominator has three factors.

The denominator has three terms.

Explanation:

The numerator has two terms. This statement is correct since 2x and 3 are the only terms here.

The denominator has three factors. This statement is correct. x^2 has two factors and 2x has on factor.

The denominator has three terms. This statement is true. The denominator has terms: x^2 , 2x and 3.

2x + 3 is not a single factor

2x + 3 in the numerator does not divide with 2x + 3 in the denominator.

Answer 2

The numerator has two terms.

The denominator has three terms.

`2x+3` is a single factor in the numerator.

Explanation:

We have the following rational expression :

`(2x+3)/(x^(2)+2x+3)`

In which

`2x+3` is the numerator and

`x^(2)+2x+3` is the denominator

In any mathematical expression the symbols ''+'' and ''-'' divide the expression into terms. For example :

`1+2-3` has three terms :

1 , 2 and 3

For a polynomial P(x) a factor is any polinomial which divides evenly into P(x).

For example :

`x^(2)-1=(x+1).(x-1)`

This means that If we divide
`x^(2)-1` by
`x+1` we obtain
`x-1`

If we divide
`x^(2)-1` by
`x-1` we obtain
`x+1`

We conclude that
`x+1` and
`x-1` are factors of the expression
`x^(2)-1`. The expression
`x^(2)-1` has two factors.

Let's evaluate the statements for this rational expression.

The numerator has two terms. This is actually true. The terms are
`2x` and
`3`

The denominator has three factors. This is wrong. The degree of the denominator is two .This means that it can have at most two factors (real or imaginary).

The denominator has three terms. This is true. The terms are
`x^(2)` ,
`2x` and
`3`

`2x+3` is a single factor in the numerator. This is true because we can write the following expression :

`2x+3=2(x+(3)/(2))` and the expression
`2x+3` will be a single factor.

`2x+3` in the numerator divides out with
`2x+3` in the denominator. This is wrong. We can't forget about
`x^(2)` in the denominator If we want to make the division.