Final answer:
To divide X⁴ - 9X³ - 5X² - 3X⁴ by X³, you would combine like terms, then divide the leading term of the dividend by the leading term of the divisor and proceed with polynomial long division step by step. However, there is an apparent typo in the question with two terms for X⁴, and it's advisable to verify the problem for accuracy.
Step-by-step explanation:
To divide X⁴ - 9X³ - 5X² - 3X⁴ by X³ using long division, we first rewrite the expression by combining like terms. Since the question includes a typo with two terms for X⁴, we'll assume that the first term should be X³ or another exponent lower than 4, but for the sake of answering the question as presented, we'll use X⁴ as given. We proceed with the division:
- List the terms of the dividend (X⁴ - 9X³ - 5X² - 3X⁴) in descending order, and represent any missing terms with a zero coefficient.
- Divide the leading term of the dividend (X⁴) by the leading term of the divisor (X³), which gives us X¹ (X).
- Multiply the divisor (X³) by the result of the division (X), and subtract that from the dividend.
- Bring down the next term and repeat the steps until all terms have been divided.
Without any further context or clarification on the typo, we would have to assume that the correct polynomial division would look like dividing (X´ - 3X⁴) by X³, which simplified would be (X - 3) based on the terms provided. However, as there may be an error in the original polynomial expression, I advise double-checking the problem for accuracy.