Question

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

Check all that apply.

Answer 1

A, D and E are correct

given ( x - 4 ) is a factor then x =  4 is a root

the remainder on division by (x - 4 ) = 0 as indicated by the 0 on the right side of the quotient

(x - 4 ) is a factor of 3x² - 13x + 4 → A

the number 4is a root of f(x) = 3x² - 13x + 4 → D ( explained above )

thus 3x² - 13x + 4 ÷ (x - 4 ) = 3x - 1 → E

the quotient line 3 - 1 0

3 and - 1 are the coefficients of the linear quotient and 0 is the remainder

Answer 2

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

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,

$\begin{aligned}b\left){\vphantom{1{\underline {\begin{array}{*{20}{c}}3&{ - 13}&4\\{  }&{12}&{ - 4}\end{array}} }}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{{\underline {\begin{array}{*{20}{c}}&{a}_n&{ a_{n-1}}&_\cdot_\cdot_\cdot{a_0}\\{    }&{}&{ }\end{array}} }}} \hfill \\\begin{array}{*{20}{c}}{{\text{    }}{c}_n}&_\cdot_\cdot_\cdot{ c_0}&{{\text{    }}0}\end{array} \hfill\\\end{aligned}$

 

Here, the monic polynomial   is divided by the polynomial   that provides the polynomial   after division that is also a factor of the polynomial  .

Given:

The synthetic division is given below.

$\begin{aligned}4\left){\vphantom{1{\underline {\begin{array}{*{20}{c}}3&{ - 13}&4\\{  }&{12}&{ - 4}\end{array}} }}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{{\underline {\begin{array}{*{20}{c}}&3&{ - 13}&4\\{    }&{12}&{ - 4}\end{array}} }}} \hfill \\\begin{array}{*{20}{c}}{{\text{    }}3}&{ - 1}&{{\text{    }}0}\end{array} \hfill\\\end{aligned}$  

Step by step explanation:

We have to determine the answer among all the options.

Option A: $\left( {x - 4} \right)$ is a factor of $3{x^2} - 13x + 4$.

It can be observed from the given synthetic division the polynomial is $g\left( x \right) = \left( {x - 4} \right)$ that is divisible by the polynomial $F\left( x \right) = 3{x^2} - 13x + 4$.

Therefore, the polynomial $g\left( x \right) = \left( {x - 4} \right)$ is a factor of the polynomial $F\left( x \right) = 3{x^2} - 13x + 4$.

Therefore, the option A is correct option.

Option B: $\left( {x + 4} \right)$ is a factor of $3{x^2} - 13x + 4$.

From the option A, it has been proved that $g\left( x \right) = \left( {x - 4} \right)$ is a factor of the polynomial  

$F\left( x \right) = 3{x^2} - 13x + 4$

Therefore, $(x+4)$ is a not factor of $3{x^2} - 13x + 4$.

Thus, the option B is not correct option.

Option C: The number $-4$ is a root of  $F\left( x \right) = 3{x^2} - 13x + 4$.

From the option A, it has been proved that $g\left( x \right) = \left( {x - 4} \right)$ is a factor of the polynomial  

$F\left( x \right) = 3{x^2} - 13x + 4$

Now substitute 0 for $g\left( x \right)$ in the equation $g\left( x \right) = \left( {x - 4} \right)$ to find the root of the polynomial $F\left( x \right) = 3{x^2} - 13x + 4$ as,

$\begin{aligned}0&= \left( {x - 4} \right) \hfill\\x&= 4 \hfill\\\end{aligned}$  

It can be seen that the value of $x$ is 4 it means $-4$ is not a root of the polynomial $F\left( x \right) = 3{x^2} - 13x + 4$.

Therefore, the option C is not correct option.

Option D: The number $4$ is a root of $F\left( x \right) = 3{x^2} - 13x + 4$.

It can be seen that the value of $x$ is 4 in option C it means $4$ is a root of the polynomial $F\left( x \right) = 3{x^2} - 13x + 4$.

Therefore, the option D is correct option.

Option E: $\left( {3{x^2} - 13x + 4} \right) \div \left( {x - 4} \right) = \left( {3x - 1} \right)$  

The option E is also correct as $(x-4)$ is the factor of the polynomial  $F\left( x \right) = 3{x^2} - 13x + 4$.

Option F: $\left( {3{x^2} - 13x + 4} \right) \div \left( {x + 4} \right) = \left( {3x - 1} \right)$  

The option F is not correct as $\left( {x + 4} \right)$ is not the factor of the polynomial $F\left( x \right) = 3{x^2} - 13x + 4$.

Result:  

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

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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.