Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Certainly! Let's go through each of the given polynomials one by one and find [tex]\( p(0) \)[/tex], [tex]\( p(1) \)[/tex], and [tex]\( p(2) \)[/tex].
### (i) [tex]\( p(x) = x^2 - x + 1 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 0^2 - 0 + 1 = 1 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 1^2 - 1 + 1 = 1 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2^2 - 2 + 1 = 4 - 2 + 1 = 3 \][/tex]
So, the values are [tex]\( p(0) = 1 \)[/tex], [tex]\( p(1) = 1 \)[/tex], and [tex]\( p(2) = 3 \)[/tex].
### (ii) [tex]\( p(y) = 2 + y + 2y^2 - y^3 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 2 + 0 + 2 \cdot 0^2 - 0^3 = 2 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 2 + 1 + 2 \cdot 1^2 - 1^3 = 2 + 1 + 2 - 1 = 4 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2 + 2 + 2 \cdot 2^2 - 2^3 = 2 + 2 + 8 - 8 = 4 \][/tex]
So, the values are [tex]\( p(0) = 2 \)[/tex], [tex]\( p(1) = 4 \)[/tex], and [tex]\( p(2) = 4 \)[/tex].
### (iii) [tex]\( p(z) = z^3 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 0^3 = 0 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 1^3 = 1 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2^3 = 8 \][/tex]
So, the values are [tex]\( p(0) = 0 \)[/tex], [tex]\( p(1) = 1 \)[/tex], and [tex]\( p(2) = 8 \)[/tex].
### (iv) [tex]\( p(t) = (t-1)(t+1) \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = (0-1)(0+1) = -1 \times 1 = -1 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = (1-1)(1+1) = 0 \times 2 = 0 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = (2-1)(2+1) = 1 \times 3 = 3 \][/tex]
So, the values are [tex]\( p(0) = -1 \)[/tex], [tex]\( p(1) = 0 \)[/tex], and [tex]\( p(2) = 3 \)[/tex].
### (v) [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 0^2 - 3 \cdot 0 + 2 = 2 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 1^2 - 3 \cdot 1 + 2 = 1 - 3 + 2 = 0 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2^2 - 3 \cdot 2 + 2 = 4 - 6 + 2 = 0 \][/tex]
So, the values are [tex]\( p(0) = 2 \)[/tex], [tex]\( p(1) = 0 \)[/tex], and [tex]\( p(2) = 0 \)[/tex].
In summary, the calculated values are:
(i) [tex]\( p(x) = x^2 - x + 1 \)[/tex]:
[tex]\[ p(0) = 1, \quad p(1) = 1, \quad p(2) = 3 \][/tex]
(ii) [tex]\( p(y) = 2 + y + 2y^2 - y^3 \)[/tex]:
[tex]\[ p(0) = 2, \quad p(1) = 4, \quad p(2) = 4 \][/tex]
(iii) [tex]\( p(z) = z^3 \)[/tex]:
[tex]\[ p(0) = 0, \quad p(1) = 1, \quad p(2) = 8 \][/tex]
(iv) [tex]\( p(t) = (t-1)(t+1) \)[/tex]:
[tex]\[ p(0) = -1, \quad p(1) = 0, \quad p(2) = 3 \][/tex]
(v) [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]:
[tex]\[ p(0) = 2, \quad p(1) = 0, \quad p(2) = 0 \][/tex]
### (i) [tex]\( p(x) = x^2 - x + 1 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 0^2 - 0 + 1 = 1 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 1^2 - 1 + 1 = 1 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2^2 - 2 + 1 = 4 - 2 + 1 = 3 \][/tex]
So, the values are [tex]\( p(0) = 1 \)[/tex], [tex]\( p(1) = 1 \)[/tex], and [tex]\( p(2) = 3 \)[/tex].
### (ii) [tex]\( p(y) = 2 + y + 2y^2 - y^3 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 2 + 0 + 2 \cdot 0^2 - 0^3 = 2 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 2 + 1 + 2 \cdot 1^2 - 1^3 = 2 + 1 + 2 - 1 = 4 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2 + 2 + 2 \cdot 2^2 - 2^3 = 2 + 2 + 8 - 8 = 4 \][/tex]
So, the values are [tex]\( p(0) = 2 \)[/tex], [tex]\( p(1) = 4 \)[/tex], and [tex]\( p(2) = 4 \)[/tex].
### (iii) [tex]\( p(z) = z^3 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 0^3 = 0 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 1^3 = 1 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2^3 = 8 \][/tex]
So, the values are [tex]\( p(0) = 0 \)[/tex], [tex]\( p(1) = 1 \)[/tex], and [tex]\( p(2) = 8 \)[/tex].
### (iv) [tex]\( p(t) = (t-1)(t+1) \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = (0-1)(0+1) = -1 \times 1 = -1 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = (1-1)(1+1) = 0 \times 2 = 0 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = (2-1)(2+1) = 1 \times 3 = 3 \][/tex]
So, the values are [tex]\( p(0) = -1 \)[/tex], [tex]\( p(1) = 0 \)[/tex], and [tex]\( p(2) = 3 \)[/tex].
### (v) [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]
1. [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 0^2 - 3 \cdot 0 + 2 = 2 \][/tex]
2. [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = 1^2 - 3 \cdot 1 + 2 = 1 - 3 + 2 = 0 \][/tex]
3. [tex]\( p(2) \)[/tex]:
[tex]\[ p(2) = 2^2 - 3 \cdot 2 + 2 = 4 - 6 + 2 = 0 \][/tex]
So, the values are [tex]\( p(0) = 2 \)[/tex], [tex]\( p(1) = 0 \)[/tex], and [tex]\( p(2) = 0 \)[/tex].
In summary, the calculated values are:
(i) [tex]\( p(x) = x^2 - x + 1 \)[/tex]:
[tex]\[ p(0) = 1, \quad p(1) = 1, \quad p(2) = 3 \][/tex]
(ii) [tex]\( p(y) = 2 + y + 2y^2 - y^3 \)[/tex]:
[tex]\[ p(0) = 2, \quad p(1) = 4, \quad p(2) = 4 \][/tex]
(iii) [tex]\( p(z) = z^3 \)[/tex]:
[tex]\[ p(0) = 0, \quad p(1) = 1, \quad p(2) = 8 \][/tex]
(iv) [tex]\( p(t) = (t-1)(t+1) \)[/tex]:
[tex]\[ p(0) = -1, \quad p(1) = 0, \quad p(2) = 3 \][/tex]
(v) [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]:
[tex]\[ p(0) = 2, \quad p(1) = 0, \quad p(2) = 0 \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
