Let's analyze the given synthetic division and use it to determine which statements are true.
We start by writing the polynomial and using synthetic division to find if the number 2 is a root of the polynomial [tex]\(F(x) = 3x^2 - 11x + 10\)[/tex].
The synthetic division process is shown below:
[tex]\[2 \ \left|\begin{array}{ccc}
3 & -11 & 10 \\
& 6 & -10 \\
\hline
3 & -5 & 0 \\
\end{array}\right.\][/tex]
Here's how it works step-by-step:
1. Write down the coefficients of the polynomial [tex]\(F(x)\)[/tex], which are [tex]\(3, -11, 10\)[/tex].
2. Place the 2 to the left (this is our test root).
3. Bring down the first coefficient (3) directly.
4. Multiply 2 by the value just written below the line (3), giving [tex]\(2 \times 3 = 6\)[/tex]. Write this under the second coefficient (-11).
5. Add the values in the second column [tex]\((-11 + 6 = -5)\)[/tex].
6. Multiply 2 by the value just written below the line (-5), giving [tex]\(2 \times -5 = -10\)[/tex]. Write this under the third coefficient (10).
7. Add the values in the third column [tex]\((10 + -10 = 0)\)[/tex].
After performing the synthetic division, we end up with:
[tex]\[2 \ \left|\begin{array}{ccc}
3 & -11 & 10 \\
& 6 & -10 \\
\hline
3 & -5 & 0 \\
\end{array}\right.\][/tex]
So, the division results in coefficients for a new polynomial (3x - 5) and a remainder of 0.
From this, we can deduce the following:
- Since the remainder is 0, 2 is indeed a root of [tex]\(F(x) = 3x^2 - 11x + 10\)[/tex].
- Because 2 is a root, [tex]\((x - 2)\)[/tex] is a factor of [tex]\(F(x)\)[/tex].
- The quotient polynomial obtained from the synthetic division is [tex]\(3x - 5\)[/tex].
Now let's evaluate each of the given statements:
A. The number -2 is a root of [tex]\(F(x) = 3x^2 - 11x + 10\)[/tex]. → False. The root is 2, not -2.
B. The number 2 is a root of [tex]\(F(x) = 3x^2 - 11x + 10\)[/tex]. → True.
C. [tex]\((x - 2)\)[/tex] is a factor of [tex]\(3x^2 - 11x + 10\)[/tex]. → True.
D. [tex]\(\left(3x^2 - 11x + 10\right) \div (x-2) = 3x - 5.\)[/tex] → True.
E. [tex]\((x + 2)\)[/tex] is a factor of [tex]\(3x^2 - 11x + 10\)[/tex]. → False. Only [tex]\((x - 2)\)[/tex] is a factor.
F. [tex]\(\left(3x^2 - 11x + 10\right) \div (x + 2) = 3x - 5\)[/tex]. → False, since [tex]\((x + 2)\)[/tex] is not a factor.
Thus, the correct statements are:
B. The number 2 is a root of [tex]\(F(x) = 3x^2 - 11x + 10\)[/tex].
C. [tex]\((x - 2)\)[/tex] is a factor of [tex]\(3x^2 - 11x + 10\)[/tex].
D. [tex]\(\left(3x^2 - 11x + 10\right) \div (x-2) = 3x - 5\)[/tex].
Therefore, the indices of the true statements are:
[tex]\[ [2, 3, 4] \][/tex]
