(a^4 - 7a^2b^2 + kb^4) ÷ (a-3b)
I'm not sure how to divide this by long division or otherwise...
Assistance needed.
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The method is explained at
http://www.sosmath.com/algebra/factor/fac01/fac01.html
Your particular numerator will only factor easily for certain factors k, such as 6.
The denominator will not be one of the factors, so nothing cancels out. You will get a messy looking answer with a remainder term that is a ratio of a polynomial and x-3
To divide the given expression (a^4 - 7a^2b^2 + kb^4) by (a-3b) using long division, you can follow these steps:
Step 1: Arrange the terms in descending powers of a.
(a^4 - 7a^2b^2 + kb^4)
- (a-3b) ⟹ Subtracting the divisor (a-3b) from the dividend
Step 2: Start dividing the highest power of a in the dividend (a^4) by the highest power of a in the divisor (a).
- The result of this division is a^3.
Step 3: Multiply the divisor (a-3b) by the result of the previous division (a^3) to get a product.
- (a^3)(a-3b) = a^4 - 3a^3b
Step 4: Subtract the product from the original dividend.
- (a^4 - 7a^2b^2 + kb^4) - (a^4 - 3a^3b) = -7a^2b^2 + kb^4 + 3a^3b
Step 5: Repeat steps 2-4 with the new dividend (-7a^2b^2 + kb^4 + 3a^3b).
Step 6: Continue this process until there are no more terms left in the dividend or the degree of the remaining terms is less than the divisor.
However, since you mentioned that long division may not be clear to you, another method called synthetic division can be used to simplify the division process. Synthetic division is typically used when dividing by a linear divisor (in this case, a-3b).
Here's how to use synthetic division:
Step 1: Set up the expression in synthetic division format:
(a-3b) | (a^4 - 7a^2b^2 + kb^4)
Step 2: Bring down the first coefficient, which is a^4, to the bottom row.
a^4
Step 3: Multiply the divisor (a-3b) by the first number in the bottom row (a^4).
- (a^4)(a) = a^5
a^4
----------
a^5 + ...
Step 4: Add the product to the next coefficient in the top row (7a^2b^2).
- a^5 + 7a^2b^2 = a^5 + 7a^2b^2
a^4
----------
a^5 + 7a^2b^2 + ...
Step 5: Repeat steps 3-4 with the new result and the next coefficient in the top row (kb^4).
- (a^5 + 7a^2b^2)(a) = a^6 + 7a^3b^2
a^4
----------
a^6 + 7a^3b^2 + ...
Step 6: Continue this process until you reach the last coefficient (0).
----------
a^6 + 7a^3b^2 + 0
The final result after synthetic division will be a^6 + 7a^3b^2.
I hope this helps! Let me know if you have any further questions.