Answer :
To determine which of the given options is not a factor of the polynomial function [tex]\( f(x) = x^4 - 10x^2 + 9 \)[/tex], we can check by performing polynomial division or evaluating the polynomial at the roots given by each factor. The factorization method involves substituting the roots of the proposed factors into the polynomial and seeing whether the polynomial equals zero.
### Given Polynomial:
[tex]\[ f(x) = x^4 - 10x^2 + 9 \][/tex]
### Checking Factors
#### 1. [tex]\( x - 1 \)[/tex]
- Root: [tex]\( x = 1 \)[/tex]
- Substitute [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[ f(1) = 1^4 - 10(1)^2 + 9 = 1 - 10 + 9 = 0 \][/tex]
Since [tex]\( f(1) = 0 \)[/tex], [tex]\( x - 1 \)[/tex] is a factor of the polynomial.
#### 2. [tex]\( x + 3 \)[/tex]
- Root: [tex]\( x = -3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the polynomial:
[tex]\[ f(-3) = (-3)^4 - 10(-3)^2 + 9 = 81 - 90 + 9 = 0 \][/tex]
Since [tex]\( f(-3) = 0 \)[/tex], [tex]\( x + 3 \)[/tex] is a factor of the polynomial.
#### 3. [tex]\( x - 9 \)[/tex]
- Root: [tex]\( x = 9 \)[/tex]
- Substitute [tex]\( x = 9 \)[/tex] into the polynomial:
[tex]\[ f(9) = 9^4 - 10(9)^2 + 9 = 6561 - 810 + 9 \][/tex]
[tex]\[ f(9) = 6561 - 900 = 5670 \][/tex]
Since [tex]\( f(9) \neq 0 \)[/tex], [tex]\( x - 9 \)[/tex] is not a factor of the polynomial.
#### 4. [tex]\( x + 1 \)[/tex]
- Root: [tex]\( x = -1 \)[/tex]
- Substitute [tex]\( x = -1 \)[/tex] into the polynomial:
[tex]\[ f(-1) = (-1)^4 - 10(-1)^2 + 9 = 1 - 10 + 9 = 0 \][/tex]
Since [tex]\( f(-1) = 0 \)[/tex], [tex]\( x + 1 \)[/tex] is a factor of the polynomial.
### Conclusion
Upon evaluating each proposed factor, we determine that [tex]\( x - 9 \)[/tex] is not a factor of the polynomial [tex]\( f(x) = x^4 - 10x^2 + 9 \)[/tex]. Therefore, the answer choice that is not a factor of the polynomial is:
[tex]\[ \boxed{x - 9} \][/tex]
### Given Polynomial:
[tex]\[ f(x) = x^4 - 10x^2 + 9 \][/tex]
### Checking Factors
#### 1. [tex]\( x - 1 \)[/tex]
- Root: [tex]\( x = 1 \)[/tex]
- Substitute [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[ f(1) = 1^4 - 10(1)^2 + 9 = 1 - 10 + 9 = 0 \][/tex]
Since [tex]\( f(1) = 0 \)[/tex], [tex]\( x - 1 \)[/tex] is a factor of the polynomial.
#### 2. [tex]\( x + 3 \)[/tex]
- Root: [tex]\( x = -3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the polynomial:
[tex]\[ f(-3) = (-3)^4 - 10(-3)^2 + 9 = 81 - 90 + 9 = 0 \][/tex]
Since [tex]\( f(-3) = 0 \)[/tex], [tex]\( x + 3 \)[/tex] is a factor of the polynomial.
#### 3. [tex]\( x - 9 \)[/tex]
- Root: [tex]\( x = 9 \)[/tex]
- Substitute [tex]\( x = 9 \)[/tex] into the polynomial:
[tex]\[ f(9) = 9^4 - 10(9)^2 + 9 = 6561 - 810 + 9 \][/tex]
[tex]\[ f(9) = 6561 - 900 = 5670 \][/tex]
Since [tex]\( f(9) \neq 0 \)[/tex], [tex]\( x - 9 \)[/tex] is not a factor of the polynomial.
#### 4. [tex]\( x + 1 \)[/tex]
- Root: [tex]\( x = -1 \)[/tex]
- Substitute [tex]\( x = -1 \)[/tex] into the polynomial:
[tex]\[ f(-1) = (-1)^4 - 10(-1)^2 + 9 = 1 - 10 + 9 = 0 \][/tex]
Since [tex]\( f(-1) = 0 \)[/tex], [tex]\( x + 1 \)[/tex] is a factor of the polynomial.
### Conclusion
Upon evaluating each proposed factor, we determine that [tex]\( x - 9 \)[/tex] is not a factor of the polynomial [tex]\( f(x) = x^4 - 10x^2 + 9 \)[/tex]. Therefore, the answer choice that is not a factor of the polynomial is:
[tex]\[ \boxed{x - 9} \][/tex]
