6. Divide \( x^{3}-16 x^{2}+15 x+10 \) by \( (x-2) \). a. \( x^{2}-18 x+43, \mathrm{R} 36 \) c. \( -x^{2}+14 x+13 \) b. \( -x^{2}+18 x-43 \) d. \( x^{2}-14 x-13, \mathrm{R}-16 \) 7. Divide \( -8 x^{3}+940+12 x \) by \( (x-5) \). a. \( -8 x^{2}-40 x-188 \) c. \( 8 x^{2}+40 x+188 \) b. \( -8 x^{2}+40 x+212 \), R 1880 d. \( 8 x^{2}-40 x-212, \mathrm{R}-1880 \) 8. Use synthetic division to find \( P(2) \) for \( P(x)=x^{4}-8 x^{3}+9 x^{2}+5 x+8 \). a. 2 b. 6 c. 22 d. 26 9. Use synthetic division to find \( P(-4) \) for \( P(x)=x^{4}+7 x^{3}+5 x+4 \). a. -4 b. -208 c. 720 d. -48

Let's solve each problem step by step and identify the correct answers from the provided options. --- ### **6. Divide \( x^{3} - 16x^{2} + 15x + 10 \) by \( (x - 2) \).** **Using Synthetic Division:** 1. **Set Up:** - Divisor root: \( x = 2 \) - Coefficients: \( 1 \) (for \( x^3 \)), \( -16 \) (for \( x^2 \)), \( 15 \) (for \( x \)), and \( 10 \) (constant term). 2. **Process:** ``` 2 | 1 -16 15 10 | 2 -28 -26 -------------------------- 1 -14 -13 -16 ``` 3. **Result:** - **Quotient:** \( x^{2} - 14x - 13 \) - **Remainder:** \( -16 \) **Correct Option:** **d. \( x^{2} - 14x - 13, \mathrm{R} -16 \)** --- ### **7. Divide \( -8x^{3} + 12x + 940 \) by \( (x - 5) \).** **Using Synthetic Division:** 1. **Set Up:** - Divisor root: \( x = 5 \) - Coefficients: \( -8 \) (for \( x^3 \)), \( 0 \) (for \( x^2 \)), \( 12 \) (for \( x \)), and \( 940 \) (constant term). 2. **Process:** ``` 5 | -8 0 12 940 | -40 -200 -940 ------------------------------ -8 -40 -188 0 ``` 3. **Result:** - **Quotient:** \( -8x^{2} - 40x - 188 \) - **Remainder:** \( 0 \) **Correct Option:** **a. \( -8x^{2} - 40x - 188 \)** --- ### **8. Use synthetic division to find \( P(2) \) for \( P(x) = x^{4} - 8x^{3} + 9x^{2} + 5x + 8 \).** **Using Synthetic Division:** 1. **Set Up:** - Value to evaluate: \( x = 2 \) - Coefficients: \( 1 \) (for \( x^4 \)), \( -8 \) (for \( x^3 \)), \( 9 \) (for \( x^2 \)), \( 5 \) (for \( x \)), and \( 8 \) (constant term). 2. **Process:** ``` 2 | 1 -8 9 5 8 | 2 -12 -6 -2 ------------------------------ 1 -6 -3 -1 6 ``` 3. **Result:** - **\( P(2) = 6 \)** **Correct Option:** **b. 6** --- ### **9. Use synthetic division to find \( P(-4) \) for \( P(x) = x^{4} + 7x^{3} + 5x + 4 \).** **Note:** The polynomial \( P(x) \) is missing the \( x^{2} \) term. It can be rewritten as \( P(x) = x^{4} + 7x^{3} + 0x^{2} + 5x + 4 \). **Using Synthetic Division:** 1. **Set Up:** - Value to evaluate: \( x = -4 \) - Coefficients: \( 1 \) (for \( x^4 \)), \( 7 \) (for \( x^3 \)), \( 0 \) (for \( x^2 \)), \( 5 \) (for \( x \)), and \( 4 \) (constant term). 2. **Process:** ``` -4 | 1 7 0 5 4 | -4 12 -48 188 --------------------------------------- 1 3 12 -43 192 ``` *(Note: There seems to be a miscalculation here. Let's correct it.)* **Correct Process:** ``` -4 | 1 7 0 5 4 | -4 -12 48 -212 ----------------------------------------- 1 3 -12 53 -208 ``` 3. **Result:** - **\( P(-4) = -208 \)** **Correct Option:** **b. -208** --- **Summary of Correct Answers:** - **Question 6:** **d** - **Question 7:** **a** - **Question 8:** **b** - **Question 9:** **b**