Answer :
Let's evaluate the given polynomials at [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 4 \)[/tex] step-by-step for each polynomial.
### 1. [tex]\( p(x) = x^3 - 7x^2 + 14x - 8 \)[/tex]
- For [tex]\( x = 1 \)[/tex]
[tex]\[ p(1) = 1^3 - 7 \cdot 1^2 + 14 \cdot 1 - 8 = 1 - 7 + 14 - 8 = 0 \][/tex]
- For [tex]\( x = 2 \)[/tex]
[tex]\[ p(2) = 2^3 - 7 \cdot 2^2 + 14 \cdot 2 - 8 = 8 - 28 + 28 - 8 = 0 \][/tex]
- For [tex]\( x = 4 \)[/tex]
[tex]\[ p(4) = 4^3 - 7 \cdot 4^2 + 14 \cdot 4 - 8 = 64 - 112 + 56 - 8 = 0 \][/tex]
Thus, the values are: [tex]\((0, 0, 0)\)[/tex].
### 2. [tex]\( p(y) = y^2 - 5y + 4 \)[/tex]
- For [tex]\( y = 1 \)[/tex]
[tex]\[ p(1) = 1^2 - 5 \cdot 1 + 4 = 1 - 5 + 4 = 0 \][/tex]
- For [tex]\( y = 2 \)[/tex]
[tex]\[ p(2) = 2^2 - 5 \cdot 2 + 4 = 4 - 10 + 4 = -2 \][/tex]
- For [tex]\( y = 4 \)[/tex]
[tex]\[ p(4) = 4^2 - 5 \cdot 4 + 4 = 16 - 20 + 4 = 0 \][/tex]
Thus, the values are: [tex]\((0, -2, 0)\)[/tex].
### 3. [tex]\( p(t) = t^2 - 6t + 8 \)[/tex]
- For [tex]\( t = 1 \)[/tex]
[tex]\[ p(1) = 1^2 - 6 \cdot 1 + 8 = 1 - 6 + 8 = 3 \][/tex]
- For [tex]\( t = 2 \)[/tex]
[tex]\[ p(2) = 2^2 - 6 \cdot 2 + 8 = 4 - 12 + 8 = 0 \][/tex]
- For [tex]\( t = 4 \)[/tex]
[tex]\[ p(4) = 4^2 - 6 \cdot 4 + 8 = 16 - 24 + 8 = 0 \][/tex]
Thus, the values are: [tex]\((3, 0, 0)\)[/tex].
### 4. [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex]
[tex]\[ p(1) = 1^2 - 3 \cdot 1 + 2 = 1 - 3 + 2 = 0 \][/tex]
- For [tex]\( x = 2 \)[/tex]
[tex]\[ p(2) = 2^2 - 3 \cdot 2 + 2 = 4 - 6 + 2 = 0 \][/tex]
- For [tex]\( x = 4 \)[/tex]
[tex]\[ p(4) = 4^2 - 3 \cdot 4 + 2 = 16 - 12 + 2 = 6 \][/tex]
Thus, the values are: [tex]\((0, 0, 6)\)[/tex].
### 5. [tex]\( p(x) = x^3 + 9x^2 + 23x + 15 \)[/tex]
- For [tex]\( x = 1 \)[/tex]
[tex]\[ p(1) = 1^3 + 9 \cdot 1^2 + 23 \cdot 1 + 15 = 1 + 9 + 23 + 15 = 48 \][/tex]
- For [tex]\( x = 2 \)[/tex]
[tex]\[ p(2) = 2^3 + 9 \cdot 2^2 + 23 \cdot 2 + 15 = 8 + 36 + 46 + 15 = 105 \][/tex]
- For [tex]\( x = 4 \)[/tex]
[tex]\[ p(4) = 4^3 + 9 \cdot 4^2 + 23 \cdot 4 + 15 = 64 + 144 + 92 + 15 = 315 \][/tex]
Thus, the values are: [tex]\((48, 105, 315)\)[/tex].
Summarizing all of the results, we have:
1. [tex]\( p(x)=x^3-7x^2+14x-8 \)[/tex]: [tex]\((0, 0, 0)\)[/tex]
2. [tex]\( p(y)=y^2-5y+4 \)[/tex]: [tex]\((0, -2, 0)\)[/tex]
3. [tex]\( p(t)=t^2-6t+8 \)[/tex]: [tex]\((3, 0, 0)\)[/tex]
4. [tex]\( p(x)=x^2-3x+2 \)[/tex]: [tex]\((0, 0, 6)\)[/tex]
5. [tex]\( p(x)=x^3+9x^2+23x+15 \)[/tex]: [tex]\((48, 105, 315)\)[/tex]
### 1. [tex]\( p(x) = x^3 - 7x^2 + 14x - 8 \)[/tex]
- For [tex]\( x = 1 \)[/tex]
[tex]\[ p(1) = 1^3 - 7 \cdot 1^2 + 14 \cdot 1 - 8 = 1 - 7 + 14 - 8 = 0 \][/tex]
- For [tex]\( x = 2 \)[/tex]
[tex]\[ p(2) = 2^3 - 7 \cdot 2^2 + 14 \cdot 2 - 8 = 8 - 28 + 28 - 8 = 0 \][/tex]
- For [tex]\( x = 4 \)[/tex]
[tex]\[ p(4) = 4^3 - 7 \cdot 4^2 + 14 \cdot 4 - 8 = 64 - 112 + 56 - 8 = 0 \][/tex]
Thus, the values are: [tex]\((0, 0, 0)\)[/tex].
### 2. [tex]\( p(y) = y^2 - 5y + 4 \)[/tex]
- For [tex]\( y = 1 \)[/tex]
[tex]\[ p(1) = 1^2 - 5 \cdot 1 + 4 = 1 - 5 + 4 = 0 \][/tex]
- For [tex]\( y = 2 \)[/tex]
[tex]\[ p(2) = 2^2 - 5 \cdot 2 + 4 = 4 - 10 + 4 = -2 \][/tex]
- For [tex]\( y = 4 \)[/tex]
[tex]\[ p(4) = 4^2 - 5 \cdot 4 + 4 = 16 - 20 + 4 = 0 \][/tex]
Thus, the values are: [tex]\((0, -2, 0)\)[/tex].
### 3. [tex]\( p(t) = t^2 - 6t + 8 \)[/tex]
- For [tex]\( t = 1 \)[/tex]
[tex]\[ p(1) = 1^2 - 6 \cdot 1 + 8 = 1 - 6 + 8 = 3 \][/tex]
- For [tex]\( t = 2 \)[/tex]
[tex]\[ p(2) = 2^2 - 6 \cdot 2 + 8 = 4 - 12 + 8 = 0 \][/tex]
- For [tex]\( t = 4 \)[/tex]
[tex]\[ p(4) = 4^2 - 6 \cdot 4 + 8 = 16 - 24 + 8 = 0 \][/tex]
Thus, the values are: [tex]\((3, 0, 0)\)[/tex].
### 4. [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex]
[tex]\[ p(1) = 1^2 - 3 \cdot 1 + 2 = 1 - 3 + 2 = 0 \][/tex]
- For [tex]\( x = 2 \)[/tex]
[tex]\[ p(2) = 2^2 - 3 \cdot 2 + 2 = 4 - 6 + 2 = 0 \][/tex]
- For [tex]\( x = 4 \)[/tex]
[tex]\[ p(4) = 4^2 - 3 \cdot 4 + 2 = 16 - 12 + 2 = 6 \][/tex]
Thus, the values are: [tex]\((0, 0, 6)\)[/tex].
### 5. [tex]\( p(x) = x^3 + 9x^2 + 23x + 15 \)[/tex]
- For [tex]\( x = 1 \)[/tex]
[tex]\[ p(1) = 1^3 + 9 \cdot 1^2 + 23 \cdot 1 + 15 = 1 + 9 + 23 + 15 = 48 \][/tex]
- For [tex]\( x = 2 \)[/tex]
[tex]\[ p(2) = 2^3 + 9 \cdot 2^2 + 23 \cdot 2 + 15 = 8 + 36 + 46 + 15 = 105 \][/tex]
- For [tex]\( x = 4 \)[/tex]
[tex]\[ p(4) = 4^3 + 9 \cdot 4^2 + 23 \cdot 4 + 15 = 64 + 144 + 92 + 15 = 315 \][/tex]
Thus, the values are: [tex]\((48, 105, 315)\)[/tex].
Summarizing all of the results, we have:
1. [tex]\( p(x)=x^3-7x^2+14x-8 \)[/tex]: [tex]\((0, 0, 0)\)[/tex]
2. [tex]\( p(y)=y^2-5y+4 \)[/tex]: [tex]\((0, -2, 0)\)[/tex]
3. [tex]\( p(t)=t^2-6t+8 \)[/tex]: [tex]\((3, 0, 0)\)[/tex]
4. [tex]\( p(x)=x^2-3x+2 \)[/tex]: [tex]\((0, 0, 6)\)[/tex]
5. [tex]\( p(x)=x^3+9x^2+23x+15 \)[/tex]: [tex]\((48, 105, 315)\)[/tex]
