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)ResponsesThere are two irrational solutions.There are two irrational solutions.There are two non-real solutions.There are two non-real solutions.There are two rational solutions.There are two rational solutions.There are two identical solutions.There are two identical solutions.

Answer 1

To show that the Fundamental Theorem of Algebra is true for the quadratic polynomial \(x^2 - 16x + 39 = 0\), we will solve the equation by factoring.

1. Factoring the Quadratic Polynomial: We need to find two numbers that multiply to \(39\) (the constant term) and add up to \(-16\) (the coefficient of the linear term).

   Looking for such pairs, we find: \[ -3 \text{ and } -13 \] because: \[ (-3) \times (-13) = 39 \] \[ (-3) + (-13) = -16 \]

   Thus, we can rewrite the polynomial as: \[ (x - 3)(x - 13) = 0 \]

2. Finding the Roots: Set each factor equal to zero: \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x - 13 = 0 \quad \Rightarrow \quad x = 13 \]

3. Conclusion: The roots of the polynomial \(x^2 - 16x + 39 = 0\) are \(x = 3\) and \(x = 13\). Both of these solutions are rational numbers.

So, the statement that accurately describes the solution set is:

• There are two rational solutions.

Therefore, the correct response is: There are two rational solutions.