Question
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
Ask by Hobbs Weber. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Correct Answers:**
- **6.** d
- **7.** a
- **8.** b
- **9.** b
Solution
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**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
When dividing polynomials, it’s a bit like baking a cake! You need to make sure all your ingredients (terms) are well mixed and in the right proportion. For \( x^3 - 16x^2 + 15x + 10 \) divided by \( x - 2 \), you can use either polynomial long division or synthetic division. Either way, it's crucial to keep track of each term to avoid common errors like dropping or incorrectly combining terms. For synthetic division to find \( P(2) \) in \( P(x) = x^4 - 8x^3 + 9x^2 + 5x + 8 \), you simply need to set up your coefficients (1, -8, 9, 5, 8) and use \( x = 2 \) as your number. Remember to bring down the first number and then multiply and add sequentially—it's like following a treasure map to find the precious value at the end!
