Find the quotients and remainder when: 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc divided by 2a - 3b + 5c
4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
= (2a-3b+5c)^2
So, divide and you get 2a-3b+5c
Not getting unable to get
To find the quotients and remainder when dividing a polynomial by another polynomial, we can use polynomial long division. Here's how you can do it:
Step 1: Write the dividend (4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc) and divisor (2a - 3b + 5c) in the long division format.
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2a - 3b + 5c | 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
Step 2: The first term in the quotient will be the result of dividing the first term of the dividend (4a^2) by the first term of the divisor (2a). In this case, the first term of the quotient is 2a.
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2a - 3b + 5c | 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
| 2a
Step 3: Multiply the divisor (2a - 3b + 5c) by the first term of the quotient (2a). Then subtract this result from the dividend.
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2a - 3b + 5c | 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
| 2a
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-4ab + 10ac
Step 4: Bring down the next term from the dividend (-4ab + 10ac).
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2a - 3b + 5c | 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
| 2a - 3b
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-4ab + 10ac
+ 6ab - 9b^2
Step 5: Divide the new term (-4ab + 10ac + 6ab - 9b^2) by the first term of the divisor (2a), which is -2ab. The result is -2b.
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2a - 3b + 5c | 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
| 2a - 3b - 2b
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-4ab + 10ac
+ 6ab - 9b^2
- 10ac + 15bc
Step 6: Repeat steps 3-5 until there are no more terms to bring down.
Continuing the long division:
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2a - 3b + 5c | 4a^2 + 9b^2 + 25c^2 - 12ab + 20ac - 30bc
| 2a - 3b - 2b + 5c - 3c
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-4ab + 10ac
+ 6ab - 9b^2
- 10ac + 15bc
+ 15c^2 - 15bc
Step 7: At this point, there are no more terms to bring down. The quotient is the sum of all the terms we obtained along the way: 2a - 2b - 3c. The remainder is the final result in the division, which, in this case, is 15c^2.
Therefore, the quotient is 2a - 2b - 3c, and the remainder is 15c^2.