Question: Consider the following polynomial function. f(x) = x^4 + x^3 - 7x^2 - 5x + 10 Step 1 of 4: Factor…

Consider the following polynomial function.

f(x) = x^4 + x^3 - 7x^2 - 5x + 10

Step 1 of 4: Factor the polynomial completely.

f(x) =

Solution

To factor the polynomial \( f(x) = x^4 + x^3 - 7x^2 - 5x + 10 \) completely, follow these steps. Find possible rational roots using the Rational Root Theorem. \[ f(x) = x^4 + x^3 - 7x^2 - 5x + 10 \] Possible roots: \( \pm1, \pm2, \pm5, \pm10 \) Test \( x = 1 \): \[ f(1) = 1 + 1 - 7 - 5 + 10 = 0 \] Since \( f(1) = 0 \), \( (x - 1) \) is a factor. \[ f(x) = (x - 1)(x^3 + 2x^2 - 5x - 10) \] Factor \( x^3 + 2x^2 - 5x - 10 \) by finding its roots. Possible roots: \( \pm1, \pm2, \pm5, \pm10 \) Test \( x = -2 \): \[ f(-2) = (-2)^3 + 2(-2)^2 - 5(-2) - 10 = -8 + 8 + 10 - 10 = 0 \] Since \( f(-2) = 0 \), \( (x + 2) \) is a factor. \[ x^3 + 2x^2 - 5x - 10 = (x + 2)(x^2 - 5) \] Factor \( x^2 - 5 \): \[ x^2 - 5 = (x - \sqrt{5})(x + \sqrt{5}) \] Thus, the polynomial factors completely as: \[ f(x) = (x - 1)(x + 2)(x - \sqrt{5})(x + \sqrt{5}) \]