Answer :
To solve the problem of dividing the polynomial [tex]\(2x^4 + 5x^3 - 2x - 8\)[/tex] by [tex]\(x + 3\)[/tex], we'll perform polynomial long division. Let's go through the process step by step to find the quotient and remainder.
### Step 1: Set up the division
We have a dividend of [tex]\(2x^4 + 5x^3 - 2x - 8\)[/tex] and a divisor of [tex]\(x + 3\)[/tex].
### Step 2: Start the division process
First term:
- Divide the leading term of the dividend, [tex]\(2x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(2x^3\)[/tex].
- Multiply [tex]\(2x^3\)[/tex] by the entire divisor [tex]\(x + 3\)[/tex], which results in [tex]\(2x^4 + 6x^3\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(2x^4 + 5x^3 - 2x - 8) - (2x^4 + 6x^3) = -x^3 - 2x - 8
\][/tex]
Second term:
- Divide the new leading term, [tex]\(-x^3\)[/tex], by [tex]\(x\)[/tex], resulting in [tex]\(-x^2\)[/tex].
- Multiply [tex]\(-x^2\)[/tex] by [tex]\(x + 3\)[/tex], getting [tex]\(-x^3 - 3x^2\)[/tex].
- Subtract this from [tex]\(-x^3 - 2x - 8\)[/tex]:
[tex]\[
(-x^3 - 2x - 8) - (-x^3 - 3x^2) = 3x^2 - 2x - 8
\][/tex]
Third term:
- Divide the new leading term, [tex]\(3x^2\)[/tex], by [tex]\(x\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(3x^2 + 9x\)[/tex].
- Subtract from [tex]\(3x^2 - 2x - 8\)[/tex]:
[tex]\[
(3x^2 - 2x - 8) - (3x^2 + 9x) = -11x - 8
\][/tex]
Fourth term:
- Divide the leading term, [tex]\(-11x\)[/tex], by [tex]\(x\)[/tex], getting [tex]\(-11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(-11x - 33\)[/tex].
- Subtract from [tex]\(-11x - 8\)[/tex]:
[tex]\[
(-11x - 8) - (-11x - 33) = 25
\][/tex]
### Final Result
The division yields a quotient of [tex]\(2x^3 - x^2 + 3x - 11\)[/tex] and a remainder of [tex]\(25\)[/tex].
Based on the options provided, the correct answer is:
c. [tex]\(2x^3 - x^2 + 3x - 11; 25\)[/tex]
### Step 1: Set up the division
We have a dividend of [tex]\(2x^4 + 5x^3 - 2x - 8\)[/tex] and a divisor of [tex]\(x + 3\)[/tex].
### Step 2: Start the division process
First term:
- Divide the leading term of the dividend, [tex]\(2x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(2x^3\)[/tex].
- Multiply [tex]\(2x^3\)[/tex] by the entire divisor [tex]\(x + 3\)[/tex], which results in [tex]\(2x^4 + 6x^3\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(2x^4 + 5x^3 - 2x - 8) - (2x^4 + 6x^3) = -x^3 - 2x - 8
\][/tex]
Second term:
- Divide the new leading term, [tex]\(-x^3\)[/tex], by [tex]\(x\)[/tex], resulting in [tex]\(-x^2\)[/tex].
- Multiply [tex]\(-x^2\)[/tex] by [tex]\(x + 3\)[/tex], getting [tex]\(-x^3 - 3x^2\)[/tex].
- Subtract this from [tex]\(-x^3 - 2x - 8\)[/tex]:
[tex]\[
(-x^3 - 2x - 8) - (-x^3 - 3x^2) = 3x^2 - 2x - 8
\][/tex]
Third term:
- Divide the new leading term, [tex]\(3x^2\)[/tex], by [tex]\(x\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(3x^2 + 9x\)[/tex].
- Subtract from [tex]\(3x^2 - 2x - 8\)[/tex]:
[tex]\[
(3x^2 - 2x - 8) - (3x^2 + 9x) = -11x - 8
\][/tex]
Fourth term:
- Divide the leading term, [tex]\(-11x\)[/tex], by [tex]\(x\)[/tex], getting [tex]\(-11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(-11x - 33\)[/tex].
- Subtract from [tex]\(-11x - 8\)[/tex]:
[tex]\[
(-11x - 8) - (-11x - 33) = 25
\][/tex]
### Final Result
The division yields a quotient of [tex]\(2x^3 - x^2 + 3x - 11\)[/tex] and a remainder of [tex]\(25\)[/tex].
Based on the options provided, the correct answer is:
c. [tex]\(2x^3 - x^2 + 3x - 11; 25\)[/tex]
