Question

According to the synthetic division below which of the following statements are true?

Answer 1

A is true. Thus D is also true. The result of the synthetic division shows C is true.

The appropriate choices are ...
   A. (x - 2) is a factor of 3x² - 11x + 10
   C. (3x² - 11x + 10) ÷ (x - 2) = (3x - 5)
   D. The number 2 is a root of F(x) = 3x² - 11x + 10

Answer 2

The correct options are $\boxed{{\mathbf{Option A, C, and D}}}$.

Further explanation:  

In any synthetic division, the dividend polynomial $F\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + \cdots {a_0}$ and the divisor polynomial $g\left( x \right) = x - b$ can be written as,

$\begin{aligned}b\left){\vphantom{1{\underline {\begin{array}{*{20}{c}}3&{ - 11}&{10} \\ {}&6&{ - 10}\end{array}} }}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{{\underline {\begin{array}{*{20}{c}}{a}_n&{ a}_{n-1}&_\cdot_\cdot_\cdot{a}_0\\{}&{ }\end{array}} }}}\hfill\\\begin{array}{*{20}{c}}{{\text{ }}&{c}_n}&_\cdot_\cdot_\cdot{ c}_0&{{\text{ }}&0}\end{array} \hfill\\\end{aligned}$

Here, the monic polynomial $g\left( x \right) = x - b$ is divided by the polynomial $F\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + \cdots {a_0}$ that provides the polynomial $h\left( x \right) = {c_n}{x^n} + {c_{n - 1}}{x^{n - 1}} + {c_{n - 2}}{x^{n - 2}} + \cdots {c_0}$ after division that is also a factor of the polynomial $F\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + \cdots {a_0}$.

Given:

The synthetic division is given below.

$\begin{aligned}4\left){\vphantom{1{\underline {\begin{array}{*{20}{c}}3&{ - 11}&{10} \\ {}&6&{ - 10}\end{array}} }}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{{\underline {\begin{array}{*{20}{c}}3&{ - 11}&{10}\\{}&6&{ - 10}\end{array}} }}}\hfill\\\begin{array}{*{20}{c}}{{\text{ }}&3}&{ - 5}&{{\text{ }}&0}\end{array} \hfill\\\end{aligned}$  

Step by step explanation:

We have to determine the answer among all the options.

Option A: $(x-2)$ is a factor of $3{x^2} - 11x + 10$.

It can be observed from the given synthetic division the monic polynomial is $g\left( x \right) = \left( {x - 2} \right)$ that is divisible by the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Therefore, the polynomial $g\left( x \right) = \left( {x - 2} \right)$ is a factor of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Therefore, the option A is correct option.

Option B: $\left( {3{x^2} - 11x + 10} \right) \div \left( {x + 2} \right) = \left( {3x - 5} \right)$  

The option B is not correct as $g\left( x \right) = \left( {x + 2} \right)$ is not a factor of the polynomial  

$F\left( x \right) = 3{x^2} - 11x + 10$

Option C: $\left( {3{x^2} - 11x + 10} \right) \div \left( {x - 2} \right) = \left( {3x - 5} \right)$  

It has been proved that the polynomial $g\left( x \right) = \left( {x - 2} \right)$ is a factor of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Thus, the option C is correct option.

Option D: The number $2$ is a root of $F\left( x \right) = 3{x^2} - 11x + 10$.

From the option A it has been proved that $g\left( x \right) = \left( {x - 2} \right)$ is a factor of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Now substitute 0 for $g(x)$ in the equation $g\left( x \right) = \left( {x - 2} \right)$ to find the root of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$ as,

$\begin{aligned}0&= \left( {x - 2} \right) \hfill \\x &= 2\hfill\\\end{aligned}$  

It can be seen that the value of $x$ is 2 it means $2$ is a root of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Therefore, the option D is correct option.

Option E: The number $-2$ is a root of $F\left( x \right) = 3{x^2} - 11x + 10$.

It can be seen that the value of $x$ is 2 in option D it means $-2$ is not a root of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Therefore, the option E is not correct option.

Option F: $(x+2)$ is a factor of $3{x^2} - 11x + 10$.

The option F is not correct as $(x-2)$ is not the factor of the polynomial $F\left( x \right) = 3{x^2} - 11x + 10$.

Result:  

Therefore, the correct options are $\boxed{{\mathbf{Option A, C, and D}}}$.

Learn more:  

1. Learn more about the function is graphed below brainly.com/question/9590016
2. Learn more about the symmetry for a function brainly.com/question/1286775
3. Learn more about midpoint of the segment brainly.com/question/3269852

Answer details:

Grade: Medium school

Subject: Mathematics

Chapter: Synthetic division

Keywords: Synthetic division, polynomial, monic polynomial, function, factor, real number, root , divisible, addition, remainder, quotient, divisor, dividend.