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dusya [7]
2 years ago
5

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

Mathematics
2 answers:
oksano4ka [1.4K]2 years ago
8 0

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}}}.

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.

statuscvo [17]2 years ago
5 0

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


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