Check whether g(x) is a factor of p(x) by dividing the first polynomial by the second polynomial:
(i) `p(x) = 4x^(3) + 8x + 8x^(2) +7, g(x) =2x^(2) -x+1`, (ii) `p(x) =x^(4) - 5x -2, g(x) =2-x^(2)`, (iii) `p(x) = 13x^(3) -19x^(2) + 12x +14, g(x) =2-2x +x^(2)`

Answer

To determine whether \( g(x) \) is a factor of \( p(x) \) for the given polynomials, we will perform polynomial long division for each case. If the remainder is zero, then \( g(x) \) is a factor of \( p(x) \).

### (i) \( p(x) = 4x^3 + 8x^2 + 8x + 7 \), \( g(x) = 2x^2 - x + 1 \)

1. **Divide the leading term**: Divide \( 4x^3 \) by \( 2x^2 \) to get \( 2x \).
2. **Multiply and subtract**: Multiply \( g(x) \) by \( 2x \):
\[ 2x(2x^2 - x + 1) = 4x^3 - 2x^2 + 2x \]
Subtract this from \( p(x) \):
\[ (4x^3 + 8x^2 + 8x + 7) - (4x^3 - 2x^2 + 2x) = 10x^2 + 6x + 7 \]
3. **Repeat the process**: Now divide \( 10x^2 \) by \( 2x^2 \) to get \( 5 \).
4. **Multiply and subtract**: Multiply \( g(x) \) by \( 5 \):
\[ 5(2x^2 - x + 1) = 10x^2 - 5x + 5 \]
Subtract this from the previous result:
\[ (10x^2 + 6x + 7) - (10x^2 - 5x + 5) = 11x + 2 \]
5. **Determine the remainder**: The remainder is \( 11x + 2 \), which is not zero.

**Conclusion**: \( g(x) \) is **not a factor** of \( p(x) \).

### (ii) \( p(x) = x^4 - 5x - 2 \), \( g(x) = 2 - x^2 \)

1. **Divide the leading term**: Divide \( x^4 \) by \( -x^2 \) to get \( -x^2 \).
2. **Multiply and subtract**: Multiply \( g(x) \) by \( -x^2 \):
\[ -x^2(2 - x^2) = -2x^2 + x^4 \]
Subtract this from \( p(x) \):
\[ (x^4 - 5x - 2) - (x^4 - 2x^2) = 2x^2 - 5x - 2 \]
3. **Repeat the process**: Now divide \( 2x^2 \) by \( -x^2 \) to get \( -2 \).
4. **Multiply and subtract**: Multiply \( g(x) \) by \( -2 \):
\[ -2(2 - x^2) = -4 + 2x^2 \]
Subtract this from the previous result:
\[ (2x^2 - 5x - 2) - (2x^2 - 4) = -5x + 2 \]
5. **Determine the remainder**: The remainder is \( -5x + 2 \), which is not zero.

**Conclusion**: \( g(x) \) is **not a factor** of \( p(x) \).

### (iii) \( p(x) = 13x^3 - 19x^2 + 12x + 14 \), \( g(x) = 2 - 2x + x^2 \)

1. **Divide the leading term**: Divide \( 13x^3 \) by \( x^2 \) to get \( 13x \).
2. **Multiply and subtract**: Multiply \( g(x) \) by \( 13x \):
\[ 13x(2 - 2x + x^2) = 26x - 26x^2 + 13x^3 \]
Subtract this from \( p(x) \):
\[ (13x^3 - 19x^2 + 12x + 14) - (13x^3 - 26x^2 + 26x) = 7x^2 - 14x + 14 \]
3. **Repeat the process**: Now divide \( 7x^2 \) by \( x^2 \) to get \( 7 \).
4. **Multiply and subtract**: Multiply \( g(x) \) by \( 7 \):
\[ 7(2 - 2x + x^2) = 14 - 14x + 7x^2 \]
Subtract this from the previous result:
\[ (7x^2 - 14x + 14) - (7x^2 - 14x + 14) = 0 \]
5. **Determine the remainder**: The remainder is \( 0 \).

**Conclusion**: \( g(x) \) **is a factor** of \( p(x) \).

### Summary of Results:
- (i) Not a factor
- (ii) Not a factor
- (iii) Is a factor