Answer :
To find the quotient and remainder of the polynomial division [tex]\((2x^4 + 5x^3 - 2x - 8) \div (x + 3)\)[/tex], let's perform the division step by step:
1. Set up the division: We are dividing the polynomial [tex]\(2x^4 + 5x^3 - 2x - 8\)[/tex] by [tex]\(x + 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(2x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex].
[tex]\[
\frac{2x^4}{x} = 2x^3
\][/tex]
This gives us the first term of the quotient, [tex]\(2x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(2x^3\)[/tex] by the entire divisor [tex]\(x + 3\)[/tex]:
[tex]\[
2x^3 \cdot (x + 3) = 2x^4 + 6x^3
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(2x^4 + 5x^3 - 2x - 8) - (2x^4 + 6x^3) = -x^3 - 2x - 8
\][/tex]
4. Repeat the process: Divide the new leading term [tex]\(-x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-x^3}{x} = -x^2
\][/tex]
Add [tex]\(-x^2\)[/tex] to the quotient.
5. Multiply and subtract again: Multiply [tex]\(-x^2\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-x^2 \cdot (x + 3) = -x^3 - 3x^2
\][/tex]
Subtract:
[tex]\[
(-x^3 - 2x - 8) - (-x^3 - 3x^2) = 3x^2 - 2x - 8
\][/tex]
6. Repeat: Divide [tex]\(3x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^2}{x} = 3x
\][/tex]
Add [tex]\(3x\)[/tex] to the quotient.
7. Multiply and subtract: Multiply [tex]\(3x\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
3x \cdot (x + 3) = 3x^2 + 9x
\][/tex]
Subtract:
[tex]\[
(3x^2 - 2x - 8) - (3x^2 + 9x) = -11x - 8
\][/tex]
8. Final step: Divide [tex]\(-11x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-11x}{x} = -11
\][/tex]
Add [tex]\(-11\)[/tex] to the quotient.
9. Final multiplication and subtraction: Multiply [tex]\(-11\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-11 \cdot (x + 3) = -11x - 33
\][/tex]
Subtract to find the remainder:
[tex]\[
(-11x - 8) - (-11x - 33) = 25
\][/tex]
So, the quotient is [tex]\(2x^3 - x^2 + 3x - 11\)[/tex] and the remainder is [tex]\(25\)[/tex]. From the given options, this matches:
c. [tex]\(2x^3 - x^2 + 3x - 11 ; 25\)[/tex]
The correct answer is C.
1. Set up the division: We are dividing the polynomial [tex]\(2x^4 + 5x^3 - 2x - 8\)[/tex] by [tex]\(x + 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(2x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex].
[tex]\[
\frac{2x^4}{x} = 2x^3
\][/tex]
This gives us the first term of the quotient, [tex]\(2x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(2x^3\)[/tex] by the entire divisor [tex]\(x + 3\)[/tex]:
[tex]\[
2x^3 \cdot (x + 3) = 2x^4 + 6x^3
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(2x^4 + 5x^3 - 2x - 8) - (2x^4 + 6x^3) = -x^3 - 2x - 8
\][/tex]
4. Repeat the process: Divide the new leading term [tex]\(-x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-x^3}{x} = -x^2
\][/tex]
Add [tex]\(-x^2\)[/tex] to the quotient.
5. Multiply and subtract again: Multiply [tex]\(-x^2\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-x^2 \cdot (x + 3) = -x^3 - 3x^2
\][/tex]
Subtract:
[tex]\[
(-x^3 - 2x - 8) - (-x^3 - 3x^2) = 3x^2 - 2x - 8
\][/tex]
6. Repeat: Divide [tex]\(3x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^2}{x} = 3x
\][/tex]
Add [tex]\(3x\)[/tex] to the quotient.
7. Multiply and subtract: Multiply [tex]\(3x\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
3x \cdot (x + 3) = 3x^2 + 9x
\][/tex]
Subtract:
[tex]\[
(3x^2 - 2x - 8) - (3x^2 + 9x) = -11x - 8
\][/tex]
8. Final step: Divide [tex]\(-11x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-11x}{x} = -11
\][/tex]
Add [tex]\(-11\)[/tex] to the quotient.
9. Final multiplication and subtraction: Multiply [tex]\(-11\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-11 \cdot (x + 3) = -11x - 33
\][/tex]
Subtract to find the remainder:
[tex]\[
(-11x - 8) - (-11x - 33) = 25
\][/tex]
So, the quotient is [tex]\(2x^3 - x^2 + 3x - 11\)[/tex] and the remainder is [tex]\(25\)[/tex]. From the given options, this matches:
c. [tex]\(2x^3 - x^2 + 3x - 11 ; 25\)[/tex]
The correct answer is C.
