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)%0D%0AResponses%0D%0A%0D%0AThere are two identical solutions.%0D%0AThere are two identical solutions.%0D%0A%0D%0AThere are two irrational solutions.%0D%0AThere are two irrational solutions.%0D%0A%0D%0AThere are two rational solutions.%0D%0AThere are two rational solutions.%0D%0A%0D%0AThere are two non-real solutions.

Question

GPT-4o mini · Answer

To solve the quadratic polynomial $x^2 - 16x + 39 = 0$ by factoring, we first identify the coefficients: - $a = 1$ (coefficient of $x^2$) - $b = -16$ (coefficient of $x$) - $c = 39$ (constant term) Next, we want to see if we can factor the quadratic into the form $(x - p)(x - q) = 0$, where $p$ and $q$ are the roots we need to find. 1. We find $p$ and $q$ such that: $$ p + q = 16 $$ $$ pq = 39 $$ 2. We can list pairs of factors of 39: $$ (1, 39), (3, 13) $$ The pair that adds up to 16 is $3$ and $13$. Thus we can set: $$ p = 3, \quad q = 13 $$ 3. Therefore, we can write the factorization as: $$ (x - 3)(x - 13) = 0 $$ 4. Setting each factor to zero gives us the solutions: $$ x - 3 = 0 \Rightarrow x = 3 $$ $$ x - 13 = 0 \Rightarrow x = 13 $$ Thus, we have two distinct solutions: $x = 3$ and $x = 13$. ### Conclusion The solutions are both rational numbers. Since we are asked to determine which statement accurately describes the solution set, the correct response is: **There are two rational solutions.**