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)

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.