Questions: Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). (Enter your answer P(x)=x^4-8x^3+8x^2+23x+6 x=

Solution Steps

1. Identify the polynomial: The given polynomial is \( P(x) = x^4 - 8x^3 + 8x^2 + 23x + 6 \).
2. Find potential rational zeros: Use the Rational Root Theorem to list all possible rational zeros.
3. Test potential zeros: Use synthetic division or direct substitution to test which of the potential rational zeros are actual zeros.
4. Factor the polynomial: Once a zero is found, factor the polynomial accordingly and repeat the process for the resulting polynomial.
5. Use the quadratic formula: If the polynomial reduces to a quadratic form, use the quadratic formula to find the remaining zeros.

The given polynomial is: \[ P(x) = x^4 - 8x^3 + 8x^2 + 23x + 6 \]

Using the Rational Root Theorem, the potential rational zeros are the factors of the constant term (6) divided by the factors of the leading coefficient (1). The potential rational zeros are: \[ \pm 1, \pm 2, \pm 3, \pm 6 \]

By testing these potential zeros, we find that \( x = -1 \) and \( x = 6 \) are zeros of the polynomial.

After finding \( x = -1 \) and \( x = 6 \) as zeros, we can factor the polynomial accordingly. The remaining polynomial can be factored further or solved using the quadratic formula.

The remaining polynomial after factoring out \( (x + 1) \) and \( (x - 6) \) is: \[ x^2 - 3x - 1 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for \( ax^2 + bx + c = 0 \), we get: \[ x = \frac{3 \pm \sqrt{13}}{2} \]

Final Answer