To find the zeros of the polynomial $f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7$ using the Rational Root Theorem and synthetic division, follow these steps:

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem states that any possible rational root, $\frac{p}{q}$, of a polynomial is a factor of the constant term (in this case, 7) divided by a factor of the leading coefficient (in this case, 1). So, we need the factors of 7 and 1:

Factors of 7: ±1, ±7 Factors of 1: ±1

Possible rational roots are: $\pm 1, \pm 7$

Step 2: Test the Possible Roots using Synthetic Division

Test $x = 1$:

Use synthetic division to test if 1 is a root of $f(x)$.

1. Set up the synthetic division with 1 and the coefficients of $f(x)$ (1, -8, -8, 8, 7):

1 & 1 & -8 & -8 & 8 & 7 \\ & & 1 & -7 & -15 & -7 & 0 \\ \hline & 1 & -7 & -15 & -7 & 0 \\ \end{array}$$ Since the remainder is not 0, $$x = 1$$ is not a root. #### Test $$x = -1$$: $$\begin{array}{r|rrrrr} -1 & 1 & -8 & -8 & 8 & 7 \\ & & -1 & 9 & -1 & -7 \\ \hline & 1 & -9 & 1 & 7 & 0 \\ \end{array}$$ Since the remainder is 0, $$x = -1$$ is a root. ### Step 3: Factor out $$x + 1$$ After confirming $$x = -1$$ is a root, factor out $$x + 1$$ from $$f(x)$$: $$f(x) = (x + 1)(x^3 - 9x^2 + x + 7)$$ ### Step 4: Repeat for $$x^3 - 9x^2 + x + 7$$ Continue using the Rational Root Theorem and synthetic division to find the roots of $$x^3 - 9x^2 + x + 7$$: #### Test $$x = 1$$: $$\begin{array}{r|rrrr} 1 & 1 & -9 & 1 & 7 \\ & & 1 & -8 & -7 \\ \hline & 1 & -8 & -7 & 0 \\ \end{array}$$ Since the remainder is 0, $$x = 1$$ is a root. $$x^3 - 9x^2 + x + 7 = (x - 1)(x^2 - 8x - 7)$$ ### Step 5: Factor $$x^2 - 8x - 7$$ To factor $$x^2 - 8x - 7$$, use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$: $$x = \frac{8 \pm \sqrt{64 + 28}}{2} = \frac{8 \pm \sqrt{92}}{2} = \frac{8 \pm 2\sqrt{23}}{2} = 4 \pm \sqrt{23}$$ ### Summary The roots of $$f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7$$ are: 1. $$x = -1$$ 2. $$x = 1$$ 3. $$x = 4 + \sqrt{23}$$ 4. $$x = 4 - \sqrt{23}$$ Would you like more details or have any questions? Here are some more questions you might want to ask: 1. How do you use synthetic division for polynomial division? 2. What is the Rational Root Theorem? 3. How do you factor a quadratic equation? 4. What are the steps for using the quadratic formula? 5. Can you explain another example of finding polynomial roots? Tip: Practice using synthetic division and the Rational Root Theorem with different polynomials to strengthen your understanding of the process.