Questions: Given the equation x^4+4x^3-5x^2-16x+4=0, complete the following. a. List all possible rational roots. b. Use synthetic division to test several possible rational roots in order to identify one actual root. c. Use the root from part (b) to solve the equation. a. List all rational roots that are possible according to the Rational Zero Theorem. ±1, ±2, ±4 (Use commas to separate answers as needed.) b. Use synthetic division to test several possible rational roots in order to identify one actual root. One rational root of the given equation is (Simplify your answer.)

Solution Steps

According to the Rational Root Theorem, the possible rational roots of the polynomial \(x^{4}+4x^{3}-5x^{2}-16x+4=0\) are: \[ \pm 1, \pm 2, \pm 4 \]

We tested the possible rational roots using synthetic division:

• Testing \(x = 1\):

  • Quotient: \(x^{3} + 5x^{2} - 16\)
  • Remainder: \(-12\)

• Testing \(x = -1\):

  • Quotient: \(x^{3} + 3x^{2} - 8x - 8\)
  • Remainder: \(12\)

• Testing \(x = 2\):

  • Quotient: \(x^{3} + 6x^{2} + 7x - 2\)
  • Remainder: \(0\)

From the tests, we found that one rational root of the given equation is: \[ x = 2 \]

Using the root \(x = 2\), we performed polynomial long division on \(x^{4}+4x^{3}-5x^{2}-16x+4\) by \(x - 2\). The result of the division is: \[ \frac{x^{4} + 4x^{3} - 5x^{2} - 16x + 4}{x - 2} = x^{3} + 6x^{2} + 7x - 2 \] with a remainder of \(0\).

Final Answer