Find the quotient and remainder. Show your step-by-step work. (2x^3+9x^2+11x+3)÷(2x+3)
To find the quotient and remainder of the division, we can use polynomial long division.
Step 1: Divide the first term of the dividend (2x^3) by the first term of the divisor (2x). The result is x^2.
x^2
___________
2x+3 | 2x^3 + 9x^2 + 11x + 3
- (2x^3 + 3x^2)
___________
6x^2 + 11x + 3
Step 2: Multiply the result obtained in step 1 (x^2) by the entire divisor (2x+3). Write this result under the dividend, then subtract.
x^2
___________
2x+3 | 2x^3 + 9x^2 + 11x + 3
- (2x^3 + 3x^2)
___________
6x^2 + 11x + 3
- (6x^2 + 9x)
___________
2x + 3
Step 3: Repeat steps 1 and 2 using the remaining terms.
x^2 + 2x
___________
2x+3 | 2x^3 + 9x^2 + 11x + 3
- (2x^3 + 3x^2)
___________
6x^2 + 11x + 3
- (6x^2 + 9x)
___________
2x + 3
- (2x + 3)
___________
0
Step 4: Since there are no remaining terms to divide and subtract, the process is complete. The quotient is x^2 + 2x and the remainder is 0.
Therefore, the quotient is x^2 + 2x and the remainder is 0.