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Answer :
To find which polynomial from the given options is equivalent to [tex]\( f(x) = -x^2 + 3x \)[/tex], let's review each option:
1. [tex]\(-x^2 + 3x\)[/tex] – This is the function we have, so we are looking for an expression that matches this.
2. [tex]\(-x^4 - 6x^3 - 12x^2 + 9x\)[/tex] – This does not match because it includes terms with powers greater than 2 and constants that don't match [tex]\(-x^2 + 3x\)[/tex].
3. [tex]\(-x^4 - 9x^2 + 3x\)[/tex] – This does not match because it includes a term with a power of 4 and has an extra term [tex]\(-9x^2\)[/tex] instead of [tex]\(-x^2\)[/tex].
4. [tex]\(-x^4 - 6x^3 - 9x^2 + 3x\)[/tex] – This does not match because it includes terms with higher powers that are not present in the original function.
5. [tex]\(-x^4 + 6x^3 - 12x^2 + 9x\)[/tex] – This does not match because it includes terms with powers greater than 2 and coefficients that don't match [tex]\(-x^2 + 3x\)[/tex].
None of the polynomial options provided are equivalent to [tex]\( f(x) = -x^2 + 3x \)[/tex]. Therefore, the correct answer is that none of the given polynomials are equivalent to [tex]\( f(x) \)[/tex].
1. [tex]\(-x^2 + 3x\)[/tex] – This is the function we have, so we are looking for an expression that matches this.
2. [tex]\(-x^4 - 6x^3 - 12x^2 + 9x\)[/tex] – This does not match because it includes terms with powers greater than 2 and constants that don't match [tex]\(-x^2 + 3x\)[/tex].
3. [tex]\(-x^4 - 9x^2 + 3x\)[/tex] – This does not match because it includes a term with a power of 4 and has an extra term [tex]\(-9x^2\)[/tex] instead of [tex]\(-x^2\)[/tex].
4. [tex]\(-x^4 - 6x^3 - 9x^2 + 3x\)[/tex] – This does not match because it includes terms with higher powers that are not present in the original function.
5. [tex]\(-x^4 + 6x^3 - 12x^2 + 9x\)[/tex] – This does not match because it includes terms with powers greater than 2 and coefficients that don't match [tex]\(-x^2 + 3x\)[/tex].
None of the polynomial options provided are equivalent to [tex]\( f(x) = -x^2 + 3x \)[/tex]. Therefore, the correct answer is that none of the given polynomials are equivalent to [tex]\( f(x) \)[/tex].
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