Sure, let's go through the solution step-by-step.
Given the synthetic division table:
[tex]\[
\begin{array}{rrr}
- 5 \longdiv { 2 } & 6 & -8 \\
& -10 & 20 \\
\hline 2 & -4 & 12
\end{array}
\][/tex]
We can decode the following information from it:
1. Polynomial Being Divided:
The original polynomial is [tex]\(2x^2 + 6x - 8\)[/tex].
2. Divisor:
The divisor in synthetic division was [tex]\((x - 5)\)[/tex], because we used -5 in the synthetic division.
3. Quotient and Remainder:
The quotient from synthetic division results in [tex]\(2x - 4\)[/tex] and the remainder is 12.
Let's check each statement:
Statement A. When [tex]\(x = 5\)[/tex], [tex]\(2x^2 + 6x - 8 = 12\)[/tex]:
Substitute [tex]\(x = 5\)[/tex] into the polynomial:
[tex]\[
2(5^2) + 6(5) - 8 = 2(25) + 30 - 8 = 50 + 30 - 8 = 72
\][/tex]
This statement claims that the expression equals 12, but it equals 72. Therefore, Statement A is false.
Statement B. When [tex]\(x = -5\)[/tex], [tex]\(2x^2 + 6x - 8 = 12\)[/tex]:
Substitute [tex]\(x = -5\)[/tex] into the polynomial:
[tex]\[
2((-5)^2) + 6(-5) - 8 = 2(25) - 30 - 8 = 50 - 30 - 8 = 12
\][/tex]
Since the expression equals 12, Statement B is true.
Statement C. When [tex]\((2x^2 + 6x - 8)\)[/tex] is divided by [tex]\((x-5)\)[/tex], the remainder is 12:
The remainder from the synthetic division is indeed 12. Therefore, Statement C is true.
Statement D. [tex]\((x-5)\)[/tex] is a factor of [tex]\(2x^2 + 6x - 8\)[/tex]:
For [tex]\((x-5)\)[/tex] to be a factor, the remainder must be 0 when the polynomial is divided by [tex]\((x-5)\)[/tex]. From the synthetic division result, we know the remainder is 12. Therefore, [tex]\((x-5)\)[/tex] is not a factor. Statement D is false.
Statement E. When [tex]\((2x^2 + 6x - 8)\)[/tex] is divided by [tex]\((x+5)\)[/tex], the remainder is 12:
Since we were previously informed that the remainder for the operation with [tex]\(x+5\)[/tex] also results in 12, Statement E is true.
Statement F. [tex]\((x+5)\)[/tex] is a factor of [tex]\(2x^2 + 6x - 8\)[/tex]:
For [tex]\((x+5)\)[/tex] to be a factor, the remainder must be 0 when the polynomial is divided by [tex]\((x+5)\)[/tex]. As previously stated, the remainder is 12. Therefore, [tex]\((x+5)\)[/tex] is not a factor. Statement F is false.
Thus, the true statements are:
- B: When [tex]\(x = -5\)[/tex], [tex]\(2x^2 + 6x - 8 = 12\)[/tex].
- C: When [tex]\((2x^2 + 6x - 8)\)[/tex] is divided by [tex]\((x-5)\)[/tex], the remainder is 12.
- E: When [tex]\((2x^2 + 6x - 8)\)[/tex] is divided by [tex]\((x+5)\)[/tex], the remainder is 12.
So, the correct answer is:
[tex]\[
\text{B, C, E}
\][/tex]
