Answer:12x³ - 36x² + x^5 = (6x²)(2x - 4) + x^5.
Step-by-step explanation:To divide the polynomial 12x³ - 36x² + x^5 by 6x², we can use polynomial long division. Here's the process:
Divide the first term of the dividend (12x³) by the first term of the divisor (6x²). The result is 2x.
Multiply the divisor (6x²) by the quotient (2x) to get 12x³.
Subtract the product from the dividend (12x³ - 36x² + x^5). The result is -24x² + x^5.
Repeat the process for the next term of the dividend, which is -24x². Divide it by the divisor (6x²) and get -4 as the quotient.
Multiply the divisor (6x²) by the quotient (-4) to get -24x².
Subtract the product from the remaining dividend (-24x² + x^5). The result is x^5.
The remainder is x^5, which means that 6x² is a factor of 12x³ - 36x² + x^5, but x^5 is not a factor of 6x².
Therefore, the final answer is:
12x³ - 36x² + x^5 = (6x²)(2x - 4) + x^5.