Question

the fundamental theorem of algebra quick check
show that the fundamental theorem of algebra is true for the quadratic polynomial $x^2 - 16x + 39 = 0$ through solving by factoring. which of the following statements accurately describes the solution set? (1 point)

there are two rational solutions

there are two non - real solutions.

there are two irrational solutions.

there are two identical solutions.

Explanation:

Step1: Factor the quadratic

We need to factor \(x^2 - 16x + 39 = 0\). We look for two numbers that multiply to \(39\) and add up to \(-16\) (the coefficient of \(x\)). The numbers are \(-3\) and \(-13\) because \((-3)\times(-13)=39\) and \((-3)+(-13)=-16\). So the factored form is \((x - 3)(x - 13)=0\).

Step2: Solve for \(x\)

Set each factor equal to zero:

• For \(x - 3 = 0\), we get \(x = 3\) (a rational number).
• For \(x - 13 = 0\), we get \(x = 13\) (a rational number).

So there are two rational solutions.

Answer:

There are two rational solutions