X3+3px+c divided by (x-a)2
It's a good question for class 9
To divide a polynomial by (x-a)^2, you can use the polynomial long division method. Here's how to do it step by step:
Step 1: Write the dividend (x^3 + 3px + c) and the divisor ((x - a)^2) in long division form:
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(x - a)^2 | x^3 + 3px + c
Step 2: Divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, divide x^3 by (x - a)^2 to get x.
____________________
(x - a)^2 | x^3 + 3px + c
- x^3
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(x - a)^2 | -3px + c
- x^3
Step 3: Multiply x by the divisor (x - a)^2 and place the result under the dividend.
____________________
(x - a)^2 | x^3 + 3px + c
- x^3
____________
3px + c
- (x - a)^2
Step 4: Divide the highest degree term of the new dividend (3px) by the highest degree term of the divisor (x^2) to get 3p.
____________________
(x - a)^2 | x^3 + 3px + c
- x^3
____________
3px + c
- (x - a)^2
- 3px^2
Step 5: Multiply 3p by the divisor (x - a)^2 and place the result under the previous subtraction.
____________________
(x - a)^2 | x^3 + 3px + c
- x^3
____________
3px + c
- (x - a)^2
- 3px^2
______________
cx - ac^2
Step 6: Repeat steps 4 and 5 with the new dividend (cx):
____________________
(x - a)^2 | x^3 + 3px + c
- x^3
____________
3px + c
- (x - a)^2
- 3px^2
______________
cx - ac^2
- cx^2
______________
-ac^2 - cx
Step 7: Now, we have the final dividend (-ac^2 - cx). Since it no longer has any terms with a degree higher than the divisor, we stop the division.
Therefore, the result of dividing x^3 + 3px + c by (x - a)^2 is:
Quotient: x + 3p - c/a
Remainder: (-ac^2 - cx) / (x - a)^2