(6x^3 – 12x + 10) ÷ (3x – 3)
To divide the polynomial (6x^3 - 12x + 10) by (3x - 3), we can use long division. Here's how you can do it step-by-step:
Step 1: Start by dividing the highest degree term of the dividend (numerator) by the highest degree term of the divisor (denominator). In this case, divide 6x^3 by 3x, which gives us 2x^2.
2x^2
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3x - 3 | 6x^3 - 12x + 10
Step 2: Multiply the divisor (3x - 3) by the quotient we just found (2x^2). This gives us 6x^3 - 6x^2.
2x^2
___________
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
Step 3: Subtract the result from Step 2 from the dividend. Here, we have -12x + 10 - (-6x^2) = -12x + 6x^2 + 10.
2x^2 + (0x - 0)
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3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
_______________
-12x + 6x^2 + 10
Step 4: Bring down the next term from the dividend (which is -12x). Now we have -12x + 6x^2 + 10 - (-12x) = 6x^2 + 10.
2x^2 + (0x - 0)
___________
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
_______________
-12x + 6x^2 + 10
- (-12x)
_______________
6x^2 + 10
Step 5: Divide the new leading term (6x^2) by the original divisor (3x - 3). This gives us 2x.
2x^2 + 2x
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3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
_______________
-12x + 6x^2 + 10
- (-12x)
_______________
6x^2 + 10
- (6x^2 - 6x)
_______________
4x + 10
Step 6: Finally, divide the remaining term (4x + 10) by the original divisor (3x - 3). This gives us the final quotient: 2x + 4.
2x^2 + 2x + 2
___________
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
_______________
-12x + 6x^2 + 10
- (-12x)
_______________
6x^2 + 10
- (6x^2 - 6x)
_______________
4x + 10
- (4x + 4)
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6
So, the quotient of (6x^3 - 12x + 10) divided by (3x - 3) is 2x + 4, with a remainder of 6.