Hello there. To solve this question, we'll have to remember some properties about polynomial functions.
We have to find which option contains a polynomial function of degree 2 that has two zeros at x = 4.
Since all the options are degree 2 polynomials, this means that we want to determine a polynomial function that has discriminant zero, since its two roots are equal.
We can determine the discriminant for the equation:
[tex]ax^2+bx+c=0,a\ne0[/tex]
with the following formula:
[tex]\Delta=b^2-4ac[/tex]
Plugging in the values for each of the options, we get, in the same order:
[tex]\begin{gathered} \Delta_1=8^2-4\cdot1\cdot16=64-64=0 \\ \Delta_2=(-8)^2-4\cdot1\cdot16=64-64=0 \\ \Delta_3=3^2-4\cdot1\cdot(-4)=9+16=25 \\ \Delta_4=0^2-4\cdot1\cdot(-16)=64 \end{gathered}[/tex]
But as you can see, two of them have discriminant zero, so we have to determine in fact which of them have the two roots equal to 4.
In this case, we know that these kind of polynomials are of the form:
[tex]x^2\pm2ax+a^2=0[/tex]
And the option that gives you roots at x = a is exactly:
[tex]x^2-2ax+a^2=0[/tex]
Because it can be factored as:
[tex](x-a)^2=0[/tex]
Therefore the correct answer is:
[tex]x^2-8x+16[/tex]
And it is contained in the option B.
