Question: Consider the following polynomial function. \[ f(x) = x^4 + x^3 - 11x^2 - 5x + 30 \] Step 3 of…

Consider the following polynomial function.

\[ f(x) = x^4 + x^3 - 11x^2 - 5x + 30 \]

Step 3 of 4: Find the x-intercept(s) at which \( f \) crosses the axis. Express the intercept(s) as ordered pair(s).

Answer

Select the number of x-intercept(s) at which \( f \) crosses the axis.

Selecting an option will display any text boxes needed to complete your answer.

• none
• 1
• 2
• 3
• 4

x-intercept(s) at which \( f \) crosses the axis:

Solution

Step 1 Find the x-intercepts by setting \( f(x) = 0 \). \[ f(x) = x^4 + x^3 - 11x^2 - 5x + 30 = 0 \] Step 2 Use the Rational Root Theorem to identify possible rational roots: \( \pm1, \pm2, \pm3, \pm5, \pm6, \pm10, \pm15, \pm30 \). Testing \( x = 2 \): \[ f(2) = (2)^4 + (2)^3 - 11(2)^2 - 5(2) + 30 = 0 \] Since \( f(2) = 0 \), \( x = 2 \) is a root. Step 3 Factor out \( (x - 2) \): \[ f(x) = (x - 2)(x^3 + 3x^2 - 5x - 15) \] Step 4 Find the remaining roots by factoring \( x^3 + 3x^2 - 5x - 15 = 0 \). Testing \( x = -3 \): \[ f(-3) = (-3)^3 + 3(-3)^2 - 5(-3) - 15 = 0 \] Since \( f(-3) = 0 \), \( x = -3 \) is a root. Factor out \( (x + 3) \): \[ f(x) = (x - 2)(x + 3)(x^2 - 5) \] Solve \( x^2 - 5 = 0 \): \[ x^2 = 5 \\ x = \sqrt{5}, \quad x = -\sqrt{5} \] The x-intercepts are: \[ (2, 0), \quad (-3, 0), \quad (\sqrt{5}, 0), \quad (-\sqrt{5}, 0) \]