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2. Find [tex]\( p(0), p(1) \)[/tex], and [tex]\( p(2) \)[/tex] for each of the following polynomials:

(i) [tex]\( p(x) = x^2 - x + 1 \)[/tex]

(ii) [tex]\( p(y) = 2 + y + 2y^2 - y^3 \)[/tex]

(iii) [tex]\( p(z) = z^3 \)[/tex]

(iv) [tex]\( p(t) = (t-1)(t+1) \)[/tex]

(v) [tex]\( p(x) = x^2 - 3x + 2 \)[/tex]

Sagot :

GinnyAnswer
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]
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