Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.