Show all work. Find the quotient:6x^3-x^2-7x-9/2x+3
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To find the quotient of the expression (6x^3 - x^2 - 7x - 9) divided by (2x + 3), we can use long division. Here are the steps:
Step 1: Start by dividing the highest degree term in the dividend, which is 6x^3, by the highest degree term in the divisor, which is 2x.
6x^3 / 2x = 3x^2
Step 2: Multiply the whole divisor (2x + 3) by the quotient obtained in the previous step, which is 3x^2. Write the result below the dividend, making sure the terms line up with their respective degrees.
3x^2 * (2x + 3) = 6x^3 + 9x^2
6x^3 - x^2 - 7x - 9
- (6x^3 + 9x^2)
______________________
-10x^2 - 7x
Step 3: Subtract the result obtained in the previous step from the original dividend. In this case, it is -10x^2 - 7x.
-10x^2 - 7x / 2x + 3
Step 4: Repeat the process by dividing the new term obtained from the previous step (-10x^2) by the divisor (2x), and then multiply the divisor by the quotient obtained. Write this result below the previous result.
-10x^2 / 2x = -5x
-5x * (2x + 3) = -10x^2 - 15x
6x^3 - x^2 - 7x - 9
- (6x^3 + 9x^2)
______________________
-10x^2 - 7x
- (-10x^2 - 15x)
____________________
8x - 9
Step 5: Repeat the process by dividing the new term obtained from the previous step (8x) by the divisor (2x), and then multiply the divisor by the quotient obtained. Write this result below the previous result.
8x / 2x = 4
4 * (2x + 3) = 8x + 12
6x^3 - x^2 - 7x - 9
- (6x^3 + 9x^2)
______________________
-10x^2 - 7x
- (-10x^2 - 15x)
____________________
8x - 9
- (8x + 12)
_______________
-21
Step 6: We reached a remainder of -21, and there are no more terms left in the dividend to divide. Therefore, our final quotient is: 3x^2 - 5x + 4, and the remainder is -21.
Therefore, the quotient of (6x^3 - x^2 - 7x - 9) divided by (2x + 3) is 3x^2 - 5x + 4 with a remainder of -21.