To determine which statements are true based on the given synthetic division and factorization information, we'll analyze each statement individually in the context of the polynomial [tex]\(F(x) = 4x^2 - 17x - 15\)[/tex].
The synthetic division table provided is:
[tex]\[
\begin{array}{rrr}
\hline 4 & -17 & -15 \\
& 20 & 15 \\
\hline 4 & 3 & 0
\end{array}
\][/tex]
This table indicates that when [tex]\(F(x)\)[/tex] is divided by [tex]\((x + 5)\)[/tex], the remainder is 0, meaning [tex]\(x + 5\)[/tex] is a factor of [tex]\(F(x)\)[/tex].
Given this information, let's evaluate each statement:
A. The number [tex]\(-5\)[/tex] is a root of [tex]\(F(x) = 4x^2 - 17x - 15\)[/tex].
- TRUE. Since [tex]\((x + 5)\)[/tex] is a factor, setting [tex]\(x + 5 = 0\)[/tex] gives [tex]\(x = -5\)[/tex], which means [tex]\(-5\)[/tex] is a root of the polynomial.
B. [tex]\((x - 5)\)[/tex] is a factor of [tex]\(4x^2 - 17x - 15\)[/tex].
- FALSE. The factor we determined is [tex]\((x + 5)\)[/tex], not [tex]\((x - 5)\)[/tex].
C. The number 5 is a root of [tex]\(F(x) = 4x^2 - 17x - 15\)[/tex].
- FALSE. As discussed, [tex]\(-5\)[/tex] is a root, not 5.
D. [tex]\((4x^2 - 17x - 15) \frac{k!}{n!(n-k)!}(x + 5) = (4x + 3)\)[/tex].
- FALSE. This statement involves an incorrect and undefined combination of polynomial and binomial coefficient notation. The left-hand side expression doesn't logically simplify to the given right-hand side expression.
E. [tex]\((x + 5)\)[/tex] is a factor of [tex]\(4x^2 - 17x - 15\)[/tex].
- TRUE. As confirmed by the synthetic division process, [tex]\((x + 5)\)[/tex] is indeed a factor of the polynomial.
F. [tex]\((4x^2 - 17x - 15) \frac{k!}{n!(n-k)!}(x - 5) = (4x + 3)\)[/tex].
- FALSE. Similar to statement D, this statement combines polynomial and binomial coefficient in an incorrect manner.
Thus, the only statements that are true are:
A. The number [tex]\(-5\)[/tex] is a root of [tex]\(F(x) = 4x^2 - 17x - 15\)[/tex].
E. [tex]\((x + 5)\)[/tex] is a factor of [tex]\(4x^2 - 17x - 15\)[/tex].
