Question

Show that the Fundamental Theorem of Algebra is true for the quadratic polynomial x2−16x+39=0 through solving by factoring. Which of the following statements accurately describes the solution set?(1 point)
Responses

There are two irrational solutions.
There are two irrational solutions.

There are two identical solutions.
There are two identical solutions.

There are two non-real solutions.
There are two non-real solutions.

There are two rational solutions.
There are two rational solutions.

Answer 1

To solve the quadratic polynomial \( x^2 - 16x + 39 = 0 \) by factoring, we first look for two numbers that multiply to \( 39 \) (the constant term) and add up to \( -16 \) (the coefficient of \( x \)).

The factors of \( 39 \) that could work are \( -3 \) and \( -13 \) because:

\[
-3 \times -13 = 39
\]
\[
-3 + (-13) = -16
\]

Thus, we can factor the quadratic as follows:

\[
x^2 - 16x + 39 = (x - 3)(x - 13) = 0
\]

Now we will set each factor equal to zero to solve for \( x \):

1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 13 = 0 \) → \( x = 13 \)

The solutions to the equation \( x^2 - 16x + 39 = 0 \) are \( x = 3 \) and \( x = 13 \), which are both rational numbers.

To summarize the solution set:

- There are two distinct solutions: \( 3 \) and \( 13 \)
- Both solutions are rational.

Thus, the correct response is:

**There are two rational solutions.**