To determine which factors are part of the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex], we need to test each factor and see if it divides the polynomial without leaving a remainder. Here is a step-by-step explanation for each factor:
1. Factor: [tex]\( x^2 - 2 \)[/tex]
- To check if [tex]\( x^2 - 2 \)[/tex] is a factor, divide the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] by [tex]\( x^2 - 2 \)[/tex].
- Result: [tex]\( x^2 - 2 \)[/tex] divides the polynomial without leaving a remainder.
- Conclusion: [tex]\( x^2 - 2 \)[/tex] is a factor.
2. Factor: [tex]\( x + 1 \)[/tex]
- To check if [tex]\( x + 1 \)[/tex] is a factor, divide the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] by [tex]\( x + 1 \)[/tex].
- Result: [tex]\( x + 1 \)[/tex] does not divide the polynomial without leaving a remainder.
- Conclusion: [tex]\( x + 1 \)[/tex] is not a factor.
3. Factor: [tex]\( x - 1 \)[/tex]
- To check if [tex]\( x - 1 \)[/tex] is a factor, divide the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] by [tex]\( x - 1 \)[/tex].
- Result: [tex]\( x - 1 \)[/tex] does not divide the polynomial without leaving a remainder.
- Conclusion: [tex]\( x - 1 \)[/tex] is not a factor.
4. Factor: [tex]\( x^2 \)[/tex]
- To check if [tex]\( x^2 \)[/tex] is a factor, divide the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] by [tex]\( x^2 \)[/tex].
- Result: [tex]\( x^2 \)[/tex] does not divide the polynomial without leaving a remainder.
- Conclusion: [tex]\( x^2 \)[/tex] is not a factor.
5. Factor: [tex]\( x \)[/tex]
- To check if [tex]\( x \)[/tex] is a factor, divide the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] by [tex]\( x \)[/tex].
- Result: [tex]\( x \)[/tex] divides the polynomial without leaving a remainder.
- Conclusion: [tex]\( x \)[/tex] is a factor.
6. Factor: [tex]\( x^2 + 2 \)[/tex]
- To check if [tex]\( x^2 + 2 \)[/tex] is a factor, divide the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] by [tex]\( x^2 + 2 \)[/tex].
- Result: [tex]\( x^2 + 2 \)[/tex] does not divide the polynomial without leaving a remainder.
- Conclusion: [tex]\( x^2 + 2 \)[/tex] is not a factor.
Based on this analysis, the factors of the polynomial [tex]\( -12x^5 + 15x^3 + 18x \)[/tex] are:
- [tex]\( x^2 - 2 \)[/tex]
- [tex]\( x \)[/tex]
