Determine the possible number of zeros, the number of \( x \)-intercepts, the number of turning points and the end behavior or the function. \( \begin{array}{ll}\text { 17) } 27 x^{9}+8 x^{6}-27 x^{3}-8=0 & \text { 18) } 3 x^{5}+9 x^{4}+34 x^{3}+102 x^{2}+63 x+189=0\end{array} \)

Let's analyze each polynomial function to determine the possible number of real zeros, \( x \)-intercepts, turning points, and end behavior. --- ### **17) \( f(x) = 27x^{9} + 8x^{6} - 27x^{3} - 8 \)** **a. Factoring the Polynomial:** 1. **Group Terms:** \[ 27x^9 - 27x^3 + 8x^6 - 8 = 27x^3(x^6 - 1) + 8(x^6 - 1) = (27x^3 + 8)(x^6 - 1) \] 2. **Factor Further:** \[ 27x^3 + 8 = (3x)^3 + 2^3 = (3x + 2)(9x^2 - 6x + 4) \] \[ x^6 - 1 = (x^3 - 1)(x^3 + 1) = (x - 1)(x^2 + x + 1)(x + 1)(x^2 - x + 1) \] 3. **Complete Factorization:** \[ f(x) = 3(x + \frac{2}{3})(9x^2 - 6x + 4)(x - 1)(x^2 + x + 1)(x + 1)(x^2 - x + 1) \] **b. Real Zeros and \( x \)-Intercepts:** - **Linear Factors:** - \( 3x + 2 = 0 \) → \( x = -\frac{2}{3} \) - \( x - 1 = 0 \) → \( x = 1 \) - \( x + 1 = 0 \) → \( x = -1 \) - **Quadratic Factors:** - \( 9x^2 - 6x + 4 \): Discriminant \( D = (-6)^2 - 4 \cdot 9 \cdot 4 = 36 - 144 = -108 < 0 \) → No real zeros. - \( x^2 + x + 1 \): Discriminant \( D = 1 - 4 = -3 < 0 \) → No real zeros. - \( x^2 - x + 1 \): Discriminant \( D = 1 - 4 = -3 < 0 \) → No real zeros. **Summary for \( f(x) \):** - **Real Zeros:** 3 - **\( x \)-Intercepts:** 3 - **Turning Points:** Up to \( 9 - 1 = 8 \) - **End Behavior:** - As \( x \to +\infty \), \( f(x) \to +\infty \) - As \( x \to -\infty \), \( f(x) \to -\infty \) --- ### **18) \( f(x) = 3x^{5} + 9x^{4} + 34x^{3} + 102x^{2} + 63x + 189 \)** **a. Finding Real Zeros:** 1. **Rational Root Theorem:** Possible rational roots are factors of 189 divided by factors of 3, i.e., \( \pm1, \pm3, \pm7, \pm9, \ldots \). 2. **Testing \( x = -3 \):** \[ f(-3) = 3(-3)^5 + 9(-3)^4 + 34(-3)^3 + 102(-3)^2 + 63(-3) + 189 = 0 \] So, \( x = -3 \) is a root. 3. **Polynomial Division:** Dividing \( f(x) \) by \( (x + 3) \) yields: \[ f(x) = (x + 3)(3x^4 + 34x^2 + 63) \] 4. **Analyzing \( 3x^4 + 34x^2 + 63 \):** - Let \( y = x^2 \): \[ 3y^2 + 34y + 63 = 0 \] - Discriminant \( D = 34^2 - 4 \cdot 3 \cdot 63 = 1156 - 756 = 400 \), so: \[ y = \frac{-34 \pm 20}{6} \Rightarrow y = -\frac{7}{3}, -9 \] - Since \( y = x^2 \) cannot be negative, there are no additional real zeros. **Summary for \( f(x) \):** - **Real Zeros:** 1 - **\( x \)-Intercepts:** 1 - **Turning Points:** Up to \( 5 - 1 = 4 \) - **End Behavior:** - As \( x \to +\infty \), \( f(x) \to +\infty \) - As \( x \to -\infty \), \( f(x) \to -\infty \) --- ### **Final Summary:** **Problem 17 Answer:** - **Real zeros and x-intercepts:** Exactly three. - **Turning points:** Up to eight. - **End behavior:** As \( x \) approaches \( +\infty \), \( f(x) \) tends to \( +\infty \) and as \( x \) approaches \( -\infty \), \( f(x) \) tends to \( -\infty \). **Problem 18 Answer:** - **Real zeros and x-intercepts:** Exactly one. - **Turning points:** Up to four. - **End behavior:** As \( x \) approaches \( +\infty \), \( f(x) \) tends to \( +\infty \) and as \( x \) approaches \( -\infty \), \( f(x) \) tends to \( -\infty \).