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Hint: To go about this problem we have to use a synthetic division method. In the synthetic division method the divisor must be in the form of (x - c).

Complete step-by-step answer:

Given Data –

Equation $3{{\text{x}}^7} - {{\text{x}}^6} + 31{{\text{x}}^4} + 21{\text{x + 5 }}$is to be divided by x+2

We can express x+2 as x-(-2) and proceed further.

First we have to set up our synthetic division, the divisor (what you are dividing by) goes outside the box and the dividend (what you are dividing into) goes inside the box.

\[ \Rightarrow {\text{ x - ( - 2)}}\left| {3{{\text{x}}^7} - {{\text{x}}^6} + 31{{\text{x}}^4} + 21{\text{x + 5 }}} \right.\]

Now write the value of c as the divisor and coefficients of the terms in the equation as dividend. Insert 0’s as coefficients for missing terms in the dividend.

$ \Rightarrow {\text{ - 2 }}\left| {{\text{ }}3{\text{ - 1 0 31 0 0 21 5}}} \right.$

Now bring down the leading coefficient (i.e. 3) to the bottom row

-2 | 3 -1 0 31 0 0 21 5

_____________________

3

Multiply c with the value just written on the bottom row, write this value right below the next coefficient in the dividend

-2 | 3 -1 0 31 0 0 21 5

-6

________________________

3

Now write their sum in the bottom row

-2 | 3 -1 0 31 0 0 21 5

-6

________________________

3 -7

Repeat the same process until the end

-2 | 3 -1 0 31 0 0 21 5

-6 14 -28 -6 12 -24 6

_____________________________

3 -7 14 3 -6 12 -3 11

The numbers in the last row represent the coefficients of the quotient and the remainder. The degree of the first term of the quotient is one less than the dividend, as the degree of the dividend 7, the degree of the first term of the quotient is 6.

Hence the quotient is $3{{\text{x}}^6} - 7{{\text{x}}^5} + 14{{\text{x}}^4} + 3{{\text{x}}^3} - 6{{\text{x}}^2} + 12{\text{x - 3}}$ and the remainder is 11.

Note –

For the questions of this type, first figure out the degree of the given equation to predict the maximum degree of the quotient. Then with help of the c value start performing a step by step synthetic division. To verify your answer, multiply the obtained quotient with the given divisor, then add the obtained remainder to it, you are supposed to get the equation given in the question.

Complete step-by-step answer:

Given Data –

Equation $3{{\text{x}}^7} - {{\text{x}}^6} + 31{{\text{x}}^4} + 21{\text{x + 5 }}$is to be divided by x+2

We can express x+2 as x-(-2) and proceed further.

First we have to set up our synthetic division, the divisor (what you are dividing by) goes outside the box and the dividend (what you are dividing into) goes inside the box.

\[ \Rightarrow {\text{ x - ( - 2)}}\left| {3{{\text{x}}^7} - {{\text{x}}^6} + 31{{\text{x}}^4} + 21{\text{x + 5 }}} \right.\]

Now write the value of c as the divisor and coefficients of the terms in the equation as dividend. Insert 0’s as coefficients for missing terms in the dividend.

$ \Rightarrow {\text{ - 2 }}\left| {{\text{ }}3{\text{ - 1 0 31 0 0 21 5}}} \right.$

Now bring down the leading coefficient (i.e. 3) to the bottom row

-2 | 3 -1 0 31 0 0 21 5

_____________________

3

Multiply c with the value just written on the bottom row, write this value right below the next coefficient in the dividend

-2 | 3 -1 0 31 0 0 21 5

-6

________________________

3

Now write their sum in the bottom row

-2 | 3 -1 0 31 0 0 21 5

-6

________________________

3 -7

Repeat the same process until the end

-2 | 3 -1 0 31 0 0 21 5

-6 14 -28 -6 12 -24 6

_____________________________

3 -7 14 3 -6 12 -3 11

The numbers in the last row represent the coefficients of the quotient and the remainder. The degree of the first term of the quotient is one less than the dividend, as the degree of the dividend 7, the degree of the first term of the quotient is 6.

Hence the quotient is $3{{\text{x}}^6} - 7{{\text{x}}^5} + 14{{\text{x}}^4} + 3{{\text{x}}^3} - 6{{\text{x}}^2} + 12{\text{x - 3}}$ and the remainder is 11.

Note –

For the questions of this type, first figure out the degree of the given equation to predict the maximum degree of the quotient. Then with help of the c value start performing a step by step synthetic division. To verify your answer, multiply the obtained quotient with the given divisor, then add the obtained remainder to it, you are supposed to get the equation given in the question.