To construct a polynomial given its zeros, you follow these steps:
1. Identify the zeros: The zeros given are [tex]\(-4, 0, 1,\)[/tex] and [tex]\(4\)[/tex].
2. Form the factors: Each zero corresponds to a factor of the polynomial. Therefore, the factors are:
[tex]\[
(x + 4), \quad x, \quad (x - 1), \quad (x - 4)
\][/tex]
3. Multiply the factors together: We need to find the product of these factors to get the polynomial in factored form:
[tex]\[
(x + 4) \cdot x \cdot (x - 1) \cdot (x - 4)
\][/tex]
4. Expand the polynomial: Now, we expand this product to write the polynomial in standard form.
Let's multiply the factors step by step.
First, multiply [tex]\((x + 4)\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[
(x + 4) \cdot x = x^2 + 4x
\][/tex]
Next, multiply [tex]\((x - 1)\)[/tex] and [tex]\((x - 4)\)[/tex]:
[tex]\[
(x - 1)(x - 4) = x^2 - 4x - x + 4 = x^2 - 5x + 4
\][/tex]
Now multiply the two results together, [tex]\((x^2 + 4x)\)[/tex] and [tex]\((x^2 - 5x + 4)\)[/tex]:
[tex]\[
(x^2 + 4x)(x^2 - 5x + 4) = x^4 - 5x^3 + 4x^2 + 4x^3 - 20x^2 + 16x
\][/tex]
Combine like terms:
[tex]\[
x^4 - x^3 - 16x^2 + 16x
\][/tex]
So, the polynomial in standard form with zeros [tex]\(-4, 0, 1,\)[/tex] and [tex]\(4\)[/tex] is:
[tex]\[
x^4 - x^3 - 16x^2 + 16x
\][/tex]
Thus, the correct answer is:
B. [tex]\(x^4 - x^3 - 16x^2 + 16x\)[/tex]
