Math / Algebra

QuestionDivide using synthetic division. $\begin{array}{l} \left(6 x^{3}-4 x^{2}+3 x-2\right) \div(x-1) \\ \left(6 x^{3}-4 x^{2}+3 x-2\right) \div(x-1)= \end{array}$ $\square$ (Simplify your answer. Do not factor.)

Studdy Solution

STEP 1

1. We are given the polynomial $6x^3 - 4x^2 + 3x - 2$ .
2. We need to divide this polynomial by $x - 1$ using synthetic division.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division process.
3. Write the quotient and remainder.
4. Simplify the final expression.

STEP 3

Identify the coefficients of the polynomial $6x^3 - 4x^2 + 3x - 2$ , which are $6, -4, 3, -2$ .
Use the root of the divisor $x - 1$ , which is $1$ , as the number for synthetic division.
Set up the synthetic division as follows:
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & & & \\ \hline & & & & \\ \end{array}$

STEP 4

Carry down the leading coefficient $6$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & & & \\ \hline & 6 & & & \\ \end{array}$
Multiply $6$ by $1$ and write the result under the next coefficient $-4$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & 6 & & \\ \hline & 6 & & & \\ \end{array}$
Add $-4$ and $6$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & 6 & & \\ \hline & 6 & 2 & & \\ \end{array}$
Multiply $2$ by $1$ and write the result under the next coefficient $3$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & 6 & 2 & \\ \hline & 6 & 2 & & \\ \end{array}$
Add $3$ and $2$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & 6 & 2 & \\ \hline & 6 & 2 & 5 & \\ \end{array}$
Multiply $5$ by $1$ and write the result under the next coefficient $-2$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & 6 & 2 & 5 \\ \hline & 6 & 2 & 5 & \\ \end{array}$
Add $-2$ and $5$ :
$\begin{array}{c|cccc} 1 & 6 & -4 & 3 & -2 \\ & & 6 & 2 & 5 \\ \hline & 6 & 2 & 5 & 3 \\ \end{array}$

STEP 5

The bottom row $6, 2, 5$ represents the coefficients of the quotient polynomial, and the last number $3$ is the remainder.
The quotient is $6x^2 + 2x + 5$ and the remainder is $3$ .

STEP 6

Write the final expression as:
$6x^2 + 2x + 5 + \frac{3}{x-1}$
The simplified answer is:
$\boxed{6x^2 + 2x + 5 + \frac{3}{x-1}}$

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QuestionDivide using synthetic division. $\begin{array}{l} \left(6 x^{3}-4 x^{2}+3 x-2\right) \div(x-1) \\ \left(6 x^{3}-4 x^{2}+3 x-2\right) \div(x-1)= \end{array}$ $\square$ (Simplify your answer. Do not factor.)

Studdy Solution

STEP 1

1. We are given the polynomial $6x^3 - 4x^2 + 3x - 2$ .
2. We need to divide this polynomial by $x - 1$ using synthetic division.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division process.
3. Write the quotient and remainder.
4. Simplify the final expression.

STEP 3

STEP 4

STEP 5

The bottom row $6, 2, 5$ represents the coefficients of the quotient polynomial, and the last number $3$ is the remainder.
The quotient is $6x^2 + 2x + 5$ and the remainder is $3$ .

STEP 6

Write the final expression as:
$6x^2 + 2x + 5 + \frac{3}{x-1}$
The simplified answer is:
$\boxed{6x^2 + 2x + 5 + \frac{3}{x-1}}$

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Studdy Solution

STEP 1

1. We are given the polynomial 6x3−4x2+3x−26x^3 - 4x^2 + 3x - 26x3−4x2+3x−2.2. We need to divide this polynomial by x−1x - 1x−1 using synthetic division.

STEP 2

1. Set up the synthetic division.2. Perform the synthetic division process.3. Write the quotient and remainder.4. Simplify the final expression.

STEP 3

STEP 4

STEP 5

The bottom row 6,2,56, 2, 56,2,5 represents the coefficients of the quotient polynomial, and the last number 333 is the remainder.The quotient is 6x2+2x+56x^2 + 2x + 56x2+2x+5 and the remainder is 333.

STEP 6

Write the final expression as:6x2+2x+5+3x−16x^2 + 2x + 5 + \frac{3}{x-1}6x2+2x+5+x−13​ The simplified answer is:6x2+2x+5+3x−1\boxed{6x^2 + 2x + 5 + \frac{3}{x-1}}6x2+2x+5+x−13​​

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1. We are given the polynomial $6x^3 - 4x^2 + 3x - 2$ .
2. We need to divide this polynomial by $x - 1$ using synthetic division.

1. Set up the synthetic division.
2. Perform the synthetic division process.
3. Write the quotient and remainder.
4. Simplify the final expression.

The bottom row $6, 2, 5$ represents the coefficients of the quotient polynomial, and the last number $3$ is the remainder.
The quotient is $6x^2 + 2x + 5$ and the remainder is $3$ .

Write the final expression as:
$6x^2 + 2x + 5 + \frac{3}{x-1}$
The simplified answer is:
$\boxed{6x^2 + 2x + 5 + \frac{3}{x-1}}$