Math / Algebra

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Use the long division method to find the result when $6 x^{3}-7 x^{2}+23 x-7$ is divided by $3 x-2$ . If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$ .
Answer Attempt 1 out of 2 $\square$ Submit Answer

Studdy Solution

STEP 1

1. We are given the polynomial $6x^3 - 7x^2 + 23x - 7$ .
2. We need to divide this polynomial by $3x - 2$ .
3. We will use the long division method.
4. If there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$ .

STEP 2

1. Set up the long division.
2. Perform the division step by step.
3. Write the quotient and remainder.
4. Express the result in the required form.

STEP 3

Set up the long division by writing $6x^3 - 7x^2 + 23x - 7$ under the division symbol and $3x - 2$ outside.

STEP 4

Divide the first term of the dividend $6x^3$ by the first term of the divisor $3x$ :
$\frac{6x^3}{3x} = 2x^2$
Write $2x^2$ as the first term of the quotient.

STEP 5

Multiply $2x^2$ by the entire divisor $3x - 2$ :
$2x^2 \times (3x - 2) = 6x^3 - 4x^2$
Subtract this product from the original polynomial:
$(6x^3 - 7x^2 + 23x - 7) - (6x^3 - 4x^2) = -3x^2 + 23x - 7$

STEP 6

Divide the first term of the new dividend $-3x^2$ by the first term of the divisor $3x$ :
$\frac{-3x^2}{3x} = -x$
Write $-x$ as the next term of the quotient.

STEP 7

Multiply $-x$ by the entire divisor $3x - 2$ :
$-x \times (3x - 2) = -3x^2 + 2x$
Subtract this product from the current dividend:
$(-3x^2 + 23x - 7) - (-3x^2 + 2x) = 21x - 7$

STEP 8

Divide the first term of the new dividend $21x$ by the first term of the divisor $3x$ :
$\frac{21x}{3x} = 7$
Write $7$ as the next term of the quotient.

STEP 9

Multiply $7$ by the entire divisor $3x - 2$ :
$7 \times (3x - 2) = 21x - 14$
Subtract this product from the current dividend:
$(21x - 7) - (21x - 14) = 7$

STEP 10

The quotient is $2x^2 - x + 7$ and the remainder is $7$ .

STEP 11

Express the result in the form $q(x) + \frac{r(x)}{b(x)}$ :
$2x^2 - x + 7 + \frac{7}{3x - 2}$
The result of the division is:
$\boxed{2x^2 - x + 7 + \frac{7}{3x - 2}}$

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QuestionQuestion Watch Video Show Examples
Use the long division method to find the result when $6 x^{3}-7 x^{2}+23 x-7$ is divided by $3 x-2$ . If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$ .
Answer Attempt 1 out of 2 $\square$ Submit Answer

Studdy Solution

STEP 1

1. We are given the polynomial $6x^3 - 7x^2 + 23x - 7$ .
2. We need to divide this polynomial by $3x - 2$ .
3. We will use the long division method.
4. If there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$ .

STEP 2

1. Set up the long division.
2. Perform the division step by step.
3. Write the quotient and remainder.
4. Express the result in the required form.

STEP 3

Set up the long division by writing $6x^3 - 7x^2 + 23x - 7$ under the division symbol and $3x - 2$ outside.

STEP 4

Divide the first term of the dividend $6x^3$ by the first term of the divisor $3x$ :
$\frac{6x^3}{3x} = 2x^2$
Write $2x^2$ as the first term of the quotient.

STEP 5

Multiply $2x^2$ by the entire divisor $3x - 2$ :
$2x^2 \times (3x - 2) = 6x^3 - 4x^2$
Subtract this product from the original polynomial:
$(6x^3 - 7x^2 + 23x - 7) - (6x^3 - 4x^2) = -3x^2 + 23x - 7$

STEP 6

Divide the first term of the new dividend $-3x^2$ by the first term of the divisor $3x$ :
$\frac{-3x^2}{3x} = -x$
Write $-x$ as the next term of the quotient.

STEP 7

Multiply $-x$ by the entire divisor $3x - 2$ :
$-x \times (3x - 2) = -3x^2 + 2x$
Subtract this product from the current dividend:
$(-3x^2 + 23x - 7) - (-3x^2 + 2x) = 21x - 7$

STEP 8

Divide the first term of the new dividend $21x$ by the first term of the divisor $3x$ :
$\frac{21x}{3x} = 7$
Write $7$ as the next term of the quotient.

STEP 9

Multiply $7$ by the entire divisor $3x - 2$ :
$7 \times (3x - 2) = 21x - 14$
Subtract this product from the current dividend:
$(21x - 7) - (21x - 14) = 7$

STEP 10

The quotient is $2x^2 - x + 7$ and the remainder is $7$ .

STEP 11

Express the result in the form $q(x) + \frac{r(x)}{b(x)}$ :
$2x^2 - x + 7 + \frac{7}{3x - 2}$
The result of the division is:
$\boxed{2x^2 - x + 7 + \frac{7}{3x - 2}}$

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Studdy Solution

STEP 1

STEP 2

1. Set up the long division.2. Perform the division step by step.3. Write the quotient and remainder.4. Express the result in the required form.

STEP 3

Set up the long division by writing 6x3−7x2+23x−7 6x^3 - 7x^2 + 23x - 7 6x3−7x2+23x−7 under the division symbol and 3x−2 3x - 2 3x−2 outside.

STEP 4

Divide the first term of the dividend 6x3 6x^3 6x3 by the first term of the divisor 3x 3x 3x:6x33x=2x2 \frac{6x^3}{3x} = 2x^2 3x6x3​=2x2Write 2x2 2x^2 2x2 as the first term of the quotient.

STEP 5

STEP 6

Divide the first term of the new dividend −3x2-3x^2−3x2 by the first term of the divisor 3x3x3x:−3x23x=−x \frac{-3x^2}{3x} = -x 3x−3x2​=−xWrite −x-x−x as the next term of the quotient.

STEP 7

STEP 8

Divide the first term of the new dividend 21x21x21x by the first term of the divisor 3x3x3x:21x3x=7 \frac{21x}{3x} = 7 3x21x​=7Write 777 as the next term of the quotient.

STEP 9

Multiply 777 by the entire divisor 3x−23x - 23x−2:7×(3x−2)=21x−14 7 \times (3x - 2) = 21x - 14 7×(3x−2)=21x−14Subtract this product from the current dividend:(21x−7)−(21x−14)=7 (21x - 7) - (21x - 14) = 7 (21x−7)−(21x−14)=7

STEP 10

The quotient is 2x2−x+72x^2 - x + 72x2−x+7 and the remainder is 777.

STEP 11

Express the result in the form q(x)+r(x)b(x) q(x) + \frac{r(x)}{b(x)} q(x)+b(x)r(x)​:2x2−x+7+73x−2 2x^2 - x + 7 + \frac{7}{3x - 2} 2x2−x+7+3x−27​ The result of the division is:2x2−x+7+73x−2 \boxed{2x^2 - x + 7 + \frac{7}{3x - 2}} 2x2−x+7+3x−27​​

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1. Set up the long division.
2. Perform the division step by step.
3. Write the quotient and remainder.
4. Express the result in the required form.

Set up the long division by writing $6x^3 - 7x^2 + 23x - 7$ under the division symbol and $3x - 2$ outside.

Divide the first term of the dividend $6x^3$ by the first term of the divisor $3x$ :
$\frac{6x^3}{3x} = 2x^2$
Write $2x^2$ as the first term of the quotient.

Divide the first term of the new dividend $-3x^2$ by the first term of the divisor $3x$ :
$\frac{-3x^2}{3x} = -x$
Write $-x$ as the next term of the quotient.

Divide the first term of the new dividend $21x$ by the first term of the divisor $3x$ :
$\frac{21x}{3x} = 7$
Write $7$ as the next term of the quotient.

Multiply $7$ by the entire divisor $3x - 2$ :
$7 \times (3x - 2) = 21x - 14$
Subtract this product from the current dividend:
$(21x - 7) - (21x - 14) = 7$

The quotient is $2x^2 - x + 7$ and the remainder is $7$ .

Express the result in the form $q(x) + \frac{r(x)}{b(x)}$ :
$2x^2 - x + 7 + \frac{7}{3x - 2}$
The result of the division is:
$\boxed{2x^2 - x + 7 + \frac{7}{3x - 2}}$