To construct a polynomial function based on the given zeros and their multiplicities, we can use the following information:
- The zero \( x = 4 \) has a multiplicity of 1, which means it contributes a factor of \( (x - 4)^1 = (x - 4) \).
- The zero \( x = -2 \) has a multiplicity of 3, which means it contributes a factor of \( (x + 2)^3 \).
Now, we can combine these factors to construct the polynomial function:
\[ f(x) = (x - 4)(x + 2)^3 \]
Next, we can expand \( (x + 2)^3 \):
\[ (x + 2)^3 = (x + 2)(x + 2)(x + 2) \]
Calculating \( (x + 2)(x + 2) \):
\[ (x + 2)(x + 2) = x^2 + 4x + 4 \]
Now multiply that by \( (x + 2) \):
\[ (x^2 + 4x + 4)(x + 2) = x^3 + 2x^2 + 4x^2 + 8x + 4 = x^3 + 6x^2 + 8x + 4 \]
So, \( (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \).
Now we can set up the full polynomial:
\[ f(x) = (x - 4)(x^3 + 6x^2 + 12x + 8) \]
Next, we can expand this as follows:
- Distributing \( (x - 4) \) to \( (x^3 + 6x^2 + 12x + 8) \):
\[ = x(x^3 + 6x^2 + 12x + 8) - 4(x^3 + 6x^2 + 12x + 8) \] \[ = x^4 + 6x^3 + 12x^2 + 8x - 4x^3 - 24x^2 - 48x - 32 \]
- Combine like terms:
\[ = x^4 + (6x^3 - 4x^3) + (12x^2 - 24x^2) + (8x - 48x) - 32 \] \[ = x^4 + 2x^3 - 12x^2 - 40x - 32 \]
So, the polynomial function Jaime should write is:
\[ f(x) = x^4 + 2x^3 - 12x^2 - 40x - 32 \]
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