Is 3 a zero of f(x) = 2x ^ 4 - 9x ^ 3 + 22x ^ 2 - 29x + 10 ? How do you know? You must show thorough , handwritten work. You may only use scratch paper , a writing utensil , and a calculator that does not have a computer algebra system.

Answer 1

To determine if \(3\) is a zero of the polynomial function \(f(x) = 2x^4 - 9x^3 + 22x^2 - 29x + 10\), we need to calculate \(f(3)\) and check if it equals \(0\).

1. Substitute \(3\) for \(x\) in the polynomial:

\[ f(3) = 2(3)^4 - 9(3)^3 + 22(3)^2 - 29(3) + 10 \]

2. Calculate each term step-by-step:

   • Calculate \(3^4\): \[ 3^4 = 81 \]
   • Calculate \(3^3\): \[ 3^3 = 27 \]
   • Calculate \(3^2\): \[ 3^2 = 9 \]

3. Now substitute these values back into the polynomial:

   • Calculate \(2(3^4)\): \[ 2 \times 81 = 162 \]
   • Calculate \(-9(3^3)\): \[ -9 \times 27 = -243 \]
   • Calculate \(22(3^2)\): \[ 22 \times 9 = 198 \]
   • Calculate \(-29(3)\): \[ -29 \times 3 = -87 \]

4. Put it all together: \[ f(3) = 162 - 243 + 198 - 87 + 10 \]

5. Calculate the sum step by step:

   • Start with \(162 - 243\): \[ 162 - 243 = -81 \]
   • Then add \(198\): \[ -81 + 198 = 117 \]
   • Then subtract \(87\): \[ 117 - 87 = 30 \]
   • Finally, add \(10\): \[ 30 + 10 = 40 \]

6. Conclusion: \[ f(3) = 40 \]

Since \(f(3) \neq 0\), \(3\) is not a zero of the function \(f(x)\).

Thus, the answer is no.