Answer :
To determine whether the given binomial is a valid factor of the polynomial, we need to perform polynomial division and check if the remainder is zero. If there's no remainder, the binomial is indeed a factor.
Let's analyze each case:
1. Polynomial: [tex]\( b^4 + 2b^3 + b^2 + 10b - 114 \)[/tex]
- Binomial: [tex]\( b + 3 \)[/tex]
- Result: False
- Explanation: When dividing the polynomial by [tex]\( b + 3 \)[/tex], the remainder is not zero.
2. Polynomial: [tex]\( x^4 + 6x^3 - 5x^2 - 32x + 40 \)[/tex]
- Binomial: [tex]\( x - 2 \)[/tex]
- Result: False
- Explanation: The remainder is not zero, so [tex]\( x - 2 \)[/tex] is not a factor.
3. Polynomial: [tex]\( 6n^5 - 16n^4 - 18n^3 + 2n^2 - 54n - 52 \)[/tex]
- Binomial: [tex]\( n - 1 \)[/tex]
- Result: False
- Explanation: Dividing by [tex]\( n - 1 \)[/tex] yields a non-zero remainder.
4. Polynomial: [tex]\( x^5 - 3x^4 - 15x^3 + 35x^2 + 54x - 72 \)[/tex]
- Binomial: [tex]\( x - 3 \)[/tex]
- Result: True
- Explanation: The division results in a zero remainder, so [tex]\( x - 3 \)[/tex] is a factor.
5. Polynomial: [tex]\( x^6 - 14x^4 + 49x^2 - 35 \)[/tex]
- Binomial: [tex]\( x - 1 \)[/tex]
- Result: False
- Explanation: The remainder is non-zero, thus [tex]\( x - 1 \)[/tex] is not a factor.
6. Polynomial: [tex]\( x^6 - 21x^4 + 80x^2 + 4x - 64 \)[/tex]
- Binomial: [tex]\( x - 5 \)[/tex]
- Result: False
- Explanation: A non-zero remainder indicates [tex]\( x - 5 \)[/tex] is not a factor.
7. Polynomial: [tex]\( 4a^4 + 8a^3 + 5a^2 + a - 18 \)[/tex]
- Binomial: [tex]\( a - 2 \)[/tex]
- Result: False
- Explanation: No zero remainder, hence [tex]\( a - 2 \)[/tex] is not a factor.
8. Polynomial: [tex]\( x^4 + 3x^3 - 21x^2 - 43x + 60 \)[/tex]
- Binomial: [tex]\( x + 5 \)[/tex]
- Result: True
- Explanation: The zero remainder confirms [tex]\( x + 5 \)[/tex] is a factor.
9. Polynomial: [tex]\( -6x^5 + 27x^2 + 54x - 31 \)[/tex]
- Binomial: [tex]\( x + 2 \)[/tex]
- Result: False
- Explanation: A non-zero remainder shows [tex]\( x + 2 \)[/tex] is not a factor.
10. Polynomial: [tex]\( x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12 \)[/tex]
- Binomial: [tex]\( x + 1 \)[/tex]
- Result: True
- Explanation: The remainder is zero, so [tex]\( x + 1 \)[/tex] is a factor.
11. Polynomial: [tex]\( 2x^6 - 6x^5 + 2x^2 - 18 \)[/tex]
- Binomial: [tex]\( x - 3 \)[/tex]
- Result: True
- Explanation: [tex]\( x - 3 \)[/tex] gives a zero remainder, making it a valid factor.
12. Polynomial: [tex]\( x^6 + 7x^5 + 7x^4 - 35x^3 - 56x^2 + 28x + 48 \)[/tex]
- Binomial: [tex]\( x + 5 \)[/tex]
- Result: False
- Explanation: The remainder isn't zero, hence [tex]\( x + 5 \)[/tex] is not a factor.
In each case where the result is "True," the given binomial is a valid factor of the polynomial, as indicated by the zero remainder upon division.
Let's analyze each case:
1. Polynomial: [tex]\( b^4 + 2b^3 + b^2 + 10b - 114 \)[/tex]
- Binomial: [tex]\( b + 3 \)[/tex]
- Result: False
- Explanation: When dividing the polynomial by [tex]\( b + 3 \)[/tex], the remainder is not zero.
2. Polynomial: [tex]\( x^4 + 6x^3 - 5x^2 - 32x + 40 \)[/tex]
- Binomial: [tex]\( x - 2 \)[/tex]
- Result: False
- Explanation: The remainder is not zero, so [tex]\( x - 2 \)[/tex] is not a factor.
3. Polynomial: [tex]\( 6n^5 - 16n^4 - 18n^3 + 2n^2 - 54n - 52 \)[/tex]
- Binomial: [tex]\( n - 1 \)[/tex]
- Result: False
- Explanation: Dividing by [tex]\( n - 1 \)[/tex] yields a non-zero remainder.
4. Polynomial: [tex]\( x^5 - 3x^4 - 15x^3 + 35x^2 + 54x - 72 \)[/tex]
- Binomial: [tex]\( x - 3 \)[/tex]
- Result: True
- Explanation: The division results in a zero remainder, so [tex]\( x - 3 \)[/tex] is a factor.
5. Polynomial: [tex]\( x^6 - 14x^4 + 49x^2 - 35 \)[/tex]
- Binomial: [tex]\( x - 1 \)[/tex]
- Result: False
- Explanation: The remainder is non-zero, thus [tex]\( x - 1 \)[/tex] is not a factor.
6. Polynomial: [tex]\( x^6 - 21x^4 + 80x^2 + 4x - 64 \)[/tex]
- Binomial: [tex]\( x - 5 \)[/tex]
- Result: False
- Explanation: A non-zero remainder indicates [tex]\( x - 5 \)[/tex] is not a factor.
7. Polynomial: [tex]\( 4a^4 + 8a^3 + 5a^2 + a - 18 \)[/tex]
- Binomial: [tex]\( a - 2 \)[/tex]
- Result: False
- Explanation: No zero remainder, hence [tex]\( a - 2 \)[/tex] is not a factor.
8. Polynomial: [tex]\( x^4 + 3x^3 - 21x^2 - 43x + 60 \)[/tex]
- Binomial: [tex]\( x + 5 \)[/tex]
- Result: True
- Explanation: The zero remainder confirms [tex]\( x + 5 \)[/tex] is a factor.
9. Polynomial: [tex]\( -6x^5 + 27x^2 + 54x - 31 \)[/tex]
- Binomial: [tex]\( x + 2 \)[/tex]
- Result: False
- Explanation: A non-zero remainder shows [tex]\( x + 2 \)[/tex] is not a factor.
10. Polynomial: [tex]\( x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12 \)[/tex]
- Binomial: [tex]\( x + 1 \)[/tex]
- Result: True
- Explanation: The remainder is zero, so [tex]\( x + 1 \)[/tex] is a factor.
11. Polynomial: [tex]\( 2x^6 - 6x^5 + 2x^2 - 18 \)[/tex]
- Binomial: [tex]\( x - 3 \)[/tex]
- Result: True
- Explanation: [tex]\( x - 3 \)[/tex] gives a zero remainder, making it a valid factor.
12. Polynomial: [tex]\( x^6 + 7x^5 + 7x^4 - 35x^3 - 56x^2 + 28x + 48 \)[/tex]
- Binomial: [tex]\( x + 5 \)[/tex]
- Result: False
- Explanation: The remainder isn't zero, hence [tex]\( x + 5 \)[/tex] is not a factor.
In each case where the result is "True," the given binomial is a valid factor of the polynomial, as indicated by the zero remainder upon division.
