Math / Algebra

QuestionWatch Video Show Examples Question
Use synthetic division to find the result when $2 x^{4}-17 x^{3}+26 x^{2}+26 x-2$ is divided by $x-6$ . If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$ . Answer Altempit out of 5

Studdy Solution

STEP 1

What is this asking? We need to divide a big polynomial by $x - 6$ using synthetic division and write the answer with any remainder. Watch out! Remember to use the root associated with the divisor, not the divisor itself, in synthetic division!
Also, keep those signs straight!

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Write the result.

STEP 3

Since we're dividing by $x - 6$ , our root is $x = 6$ .
This is what we'll use in our synthetic division setup.
Remember, it's $x - 6 = 0$ , so $x = 6$ .
Super important!

STEP 4

Now, let's grab the coefficients from our big polynomial $2x^4 - 17x^3 + 26x^2 + 26x - 2$ .
They are 2, -17, 26, 26, and -2.
Write them down in a row.

STEP 5

Draw a little "L" shape to the left of our coefficients, with the root, $6$ , on the outside.
It looks like this:
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ \end{array}$

STEP 6

Bring that first coefficient, 2, straight down below the line.
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ & 2 & & & & \end{array}$

STEP 7

Now, multiply the root, $6$ , by the number we just brought down, 2.
That gives us $6 \cdot 2 = 12$ .
Write that 12 under the next coefficient, -17.
Add those together: $-17 + 12 = -5$ .
Write the result, -5, below the line.
$\begin{array}{c c c c c c} 6 & 2 & -17 & 26 & 26 & -2 \\ \hline & 2 & -5 & & & \end{array}$ Keep going!
Multiply the root, $6$ , by -5 to get $-30$ .
Add that to 26: $26 + (-30) = -4$ .
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ 2 & -5 & -4 & & \end{array}$ Multiply $6$ by $-4$ to get $-24$ .
Add that to 26: $26 + (-24) = 2$ .
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ & 2 & -5 & -4 & 2 & \end{array}$ Finally, multiply $6$ by $2$ to get $12$ .
Add that to -2: $-2 + 12 = 10$ .
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ & 2 & -5 & -4 & 2 & 10 \end{array}$

STEP 8

Those numbers below the line are the coefficients of our quotient, and the last number is the remainder.
Since we started with an $x^4$ term and divided by an $x$ term, our quotient starts with an $x^3$ term.
So, our quotient is $2x^3 - 5x^2 - 4x + 2$ , and our remainder is 10.

STEP 9

We write the final answer as $2x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}$ .

STEP 10

$2x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}$

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QuestionWatch Video Show Examples Question
Use synthetic division to find the result when $2 x^{4}-17 x^{3}+26 x^{2}+26 x-2$ is divided by $x-6$ . If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$ . Answer Altempit out of 5

Studdy Solution

STEP 1

What is this asking? We need to divide a big polynomial by $x - 6$ using synthetic division and write the answer with any remainder. Watch out! Remember to use the root associated with the divisor, not the divisor itself, in synthetic division!
Also, keep those signs straight!

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Write the result.

STEP 3

Since we're dividing by $x - 6$ , our root is $x = 6$ .
This is what we'll use in our synthetic division setup.
Remember, it's $x - 6 = 0$ , so $x = 6$ .
Super important!

STEP 4

Now, let's grab the coefficients from our big polynomial $2x^4 - 17x^3 + 26x^2 + 26x - 2$ .
They are 2, -17, 26, 26, and -2.
Write them down in a row.

STEP 5

Draw a little "L" shape to the left of our coefficients, with the root, $6$ , on the outside.
It looks like this:
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ \end{array}$

STEP 6

Bring that first coefficient, 2, straight down below the line.
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ & 2 & & & & \end{array}$

STEP 7

STEP 8

Those numbers below the line are the coefficients of our quotient, and the last number is the remainder.
Since we started with an $x^4$ term and divided by an $x$ term, our quotient starts with an $x^3$ term.
So, our quotient is $2x^3 - 5x^2 - 4x + 2$ , and our remainder is 10.

STEP 9

We write the final answer as $2x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}$ .

STEP 10

$2x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}$

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Studdy Solution

STEP 1

What is this asking? We need to divide a big polynomial by x−6x - 6x−6 using synthetic division and write the answer with any remainder. Watch out! Remember to use the *root* associated with the divisor, not the divisor itself, in synthetic division!Also, keep those signs straight!

STEP 2

1. Set up the synthetic division.2. Perform the synthetic division.3. Write the result.

STEP 3

Since we're dividing by x−6x - 6x−6, our **root** is x=6x = 6x=6.This is what we'll use in our synthetic division setup.Remember, it's x−6=0x - 6 = 0x−6=0, so x=6x = 6x=6.Super important!

STEP 4

Now, let's grab the **coefficients** from our big polynomial 2x4−17x3+26x2+26x−22x^4 - 17x^3 + 26x^2 + 26x - 22x4−17x3+26x2+26x−2.They are **2**, **-17**, **26**, **26**, and **-2**.Write them down in a row.

STEP 5

Draw a little "L" shape to the left of our coefficients, with the **root**, 666, on the outside.It looks like this:62−172626−2\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ \end{array}6​2​−17​26​26​−2​

STEP 6

Bring that first coefficient, **2**, straight down below the line.62−172626−22\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ & 2 & & & & \end{array}6​22​−17​26​26​−2​

STEP 7

STEP 8

STEP 9

We write the final answer as 2x3−5x2−4x+2+10x−62x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}2x3−5x2−4x+2+x−610​.

STEP 10

2x3−5x2−4x+2+10x−62x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}2x3−5x2−4x+2+x−610​

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to divide a big polynomial by $x - 6$ using synthetic division and write the answer with any remainder. Watch out! Remember to use the root associated with the divisor, not the divisor itself, in synthetic division!
Also, keep those signs straight!

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Write the result.

Since we're dividing by $x - 6$ , our root is $x = 6$ .
This is what we'll use in our synthetic division setup.
Remember, it's $x - 6 = 0$ , so $x = 6$ .
Super important!

Now, let's grab the coefficients from our big polynomial $2x^4 - 17x^3 + 26x^2 + 26x - 2$ .
They are 2, -17, 26, 26, and -2.
Write them down in a row.

Draw a little "L" shape to the left of our coefficients, with the root, $6$ , on the outside.
It looks like this:
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ \end{array}$

Bring that first coefficient, 2, straight down below the line.
$\begin{array}{cccccc} 6 & 2 & -17 & 26 & 26 & -2 \\ & 2 & & & & \end{array}$

We write the final answer as $2x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}$ .

$2x^3 - 5x^2 - 4x + 2 + \frac{10}{x - 6}$