Questions: The following function is given. f(x)=x^3-2x^2-25x+50 a. List all rational zeros that are possible according to the Rational Zero Theorem.

Transcript text: The following function is given. \[ f(x)=x^{3}-2 x^{2}-25 x+50 \] a. List all rational zeros that are possible according to the Rational Zero Theorem.

Solution

Solution Steps

Step 1: Identify the constant term and leading coefficient

The given function is: \[ f(x) = x^{3} - 2x^{2} - 25x + 50 \]

The constant term (the term without \(x\)) is \(50\).
The leading coefficient (the coefficient of the highest power of \(x\)) is \(1\).

Step 2: List the factors of the constant term and leading coefficient

The factors of the constant term \(50\) are: \[ \pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50 \]
The factors of the leading coefficient \(1\) are: \[ \pm 1 \]

Step 3: Apply the Rational Zero Theorem

According to the Rational Zero Theorem, the possible rational zeros of the function are all the possible ratios of the factors of the constant term to the factors of the leading coefficient. Since the leading coefficient is \(1\), the possible rational zeros are simply the factors of the constant term: \[ \pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50 \]