Vicente was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?
Zero Multiplicity
x=16
2
x=−4
3(1 point)
Responses
f(x)=(x−16)3(x+4)2
f left parenthesis x right parenthesis equals left parenthesis x minus 16 right parenthesis cubed left parenthesis x plus 4 right parenthesis squared
f(x)=(x+16)2(x−4)3
f left parenthesis x right parenthesis equals left parenthesis x plus 16 right parenthesis squared left parenthesis x minus 4 right parenthesis cubed
f(x)=(x+16)3(x−4)2
f left parenthesis x right parenthesis equals left parenthesis x plus 16 right parenthesis cubed left parenthesis x minus 4 right parenthesis squared
f(x)=(x−16)2(x+4)3
f(x)=(x−16)2(x+4)3
Vicente should write the function:
f(x) = (x + 16)³(x - 4)²
To construct a polynomial function with given zeros and multiplicities, you need to use the factored form of the function.
The zeros are x=16 and x=-4, with multiplicities 2 and 3, respectively.
To represent a zero with multiplicity 2, you include the factor (x - 16) squared, which means (x - 16) * (x - 16).
To represent a zero with multiplicity 3, you include the factor (x + 4) cubed, which means (x + 4) * (x + 4) * (x + 4).
So the correct function would be f(x) = (x - 16)^2 * (x + 4)^3.