Studdy AI Logo
Math  /  Algebra

QuestionRonnie 23−6−510160−1030−5\begin{array}{ccccc} 2 & 3 & -6 & -5 & 10 \\ 1 & 6 & 0 & -10 \\ \hline 3 & 0 & -5 & \end{array}

Studdy Solution

STEP 1

1. We are given a polynomial 3x4−6x3−5x+10 3x^4 - 6x^3 - 5x + 10 .
2. We need to divide this polynomial by x−2 x - 2 using synthetic division.
3. The setup for synthetic division is provided, and we need to verify the steps and results.

STEP 2

1. Set up the synthetic division using the given coefficients and divisor.
2. Perform synthetic division step-by-step.
3. Verify each step of Ronnie's work.
4. Determine the quotient and remainder.

STEP 3

Identify the coefficients of the polynomial 3x4−6x3+0x2−5x+10 3x^4 - 6x^3 + 0x^2 - 5x + 10 as 3,−6,0,−5,10 3, -6, 0, -5, 10 .
The divisor is x−2 x - 2 , so the number to use in synthetic division is 2 2 .

STEP 4

Set up the synthetic division:
23−60−510\begin{array}{c|ccccc} 2 & 3 & -6 & 0 & -5 & 10 \\ & & & & & \\ \hline & & & & & \end{array}
Bring down the leading coefficient 3 3 .

STEP 5

Multiply 3 3 by 2 2 and write the result under the next coefficient:
23−60−51063\begin{array}{c|ccccc} 2 & 3 & -6 & 0 & -5 & 10 \\ & & 6 & & & \\ \hline & 3 & & & & \end{array}
Add −6 -6 and 6 6 to get 0 0 .

STEP 6

Bring down the result 0 0 , multiply by 2 2 , and write it under the next coefficient:
23−60−5106030\begin{array}{c|ccccc} 2 & 3 & -6 & 0 & -5 & 10 \\ & & 6 & 0 & & \\ \hline & 3 & 0 & & & \end{array}
Add 0 0 and 0 0 to get 0 0 .

STEP 7

Bring down the result 0 0 , multiply by 2 2 , and write it under the next coefficient:
23−60−510600300\begin{array}{c|ccccc} 2 & 3 & -6 & 0 & -5 & 10 \\ & & 6 & 0 & 0 & \\ \hline & 3 & 0 & 0 & & \end{array}
Add −5 -5 and 0 0 to get −5 -5 .

STEP 8

Bring down the result −5 -5 , multiply by 2 2 , and write it under the next coefficient:
23−60−510600−10300−5\begin{array}{c|ccccc} 2 & 3 & -6 & 0 & -5 & 10 \\ & & 6 & 0 & 0 & -10 \\ \hline & 3 & 0 & 0 & -5 & \end{array}
Add 10 10 and −10 -10 to get 0 0 .

STEP 9

Verify Ronnie's work:
The final row of the synthetic division is 3,0,0,−5,0 3, 0, 0, -5, 0 .
This indicates that the quotient is 3x3+0x2+0x−5 3x^3 + 0x^2 + 0x - 5 and the remainder is 0 0 .

STEP 10

The quotient of the division is:
3x3−5 3x^3 - 5
The remainder is 0 0 , confirming that x−2 x - 2 divides the polynomial evenly.
The solution is:
Quotient: 3x3−5 3x^3 - 5
Remainder: 0 0

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord