Question

writing polynomial functions from complex roots\which second degree polynomial function has a leading coefficient of -1 and root 4 with multiplicity 2?\\$f(x) = -x^2 - 8x + 16$\\$f(x) = -x^2 + 8x + 16$\\$f(x) = -x^2 + 8x - 16$\\$f(x) = -x^2 - 8x - 16$

Explanation:

Step1: Recall the factored form of a polynomial with a root \( r \) of multiplicity \( m \) and leading coefficient \( a \).

The general form of a second - degree polynomial (quadratic) with a root \( r \) of multiplicity 2 and leading coefficient \( a \) is \( f(x)=a(x - r)^2 \).
Here, \( a=- 1 \) and \( r = 4 \), so the polynomial should be \( f(x)=-1(x - 4)^2 \).

Step2: Expand the factored form.

First, expand \( (x - 4)^2 \) using the formula \( (a - b)^2=a^{2}-2ab + b^{2} \), where \( a=x \) and \( b = 4 \). So \( (x - 4)^2=x^{2}-8x + 16 \).
Then multiply by the leading coefficient \( - 1 \): \( f(x)=-1\times(x^{2}-8x + 16)=-x^{2}+8x - 16 \).

Answer:

\( f(x)=-x^{2}+8x - 16 \) (the second option from the left)