Divide:
3
𝑥
4
+
10
𝑥
3
−
25
𝑥
2
𝑥
+
5

Question

Divide: 
3
𝑥
4
+
10
𝑥
3
−
25
𝑥
2
𝑥
+
5

BlackTom AI · Accepted Answer

Question restatement and options:
Divide: 
3
𝑥
4
+
10
𝑥
3
−
25
𝑥
2
𝑥
+
5

Answer options:
- x
2
- 3
𝑥
2
+
5
𝑥
- 3
𝑥
3
−
5
𝑥
2
- 3
𝑥
3
+
35

Analysis of each option:
Option 1: x^2. This choice would indicate the result of the division yields a quadratic term as the leading term. If the original expression involved higher-degree terms that, when divided, reduce to x^2, this could happen, but given the provided correct answer in the prompt, this option does not match the expected outcome. It also lacks any x^3 term, which would typically appear if the dividend originally contained an x^3 term that survives the division step. Misconception to watch: assuming division always drops degree or that only even powers can appear, which isn’t generally true for polynomial divisions.

Option 2: 3x^2 + 5x. This option presents a quadratic polynomial with both x^2 and x terms but no constant or higher-degree terms. If the division were performed on a dividend that yields an x^3 term after division, you would expect an x^3 term to appear in the result, unless cancellation removes it. Since the given correct result includes an x^3 term, this option fails to account for that highest-degree term and misrepresents the degree structure of the quotient.

Option 3: 3x^3 − 5x^2. This is the correct choice according to the provided answer. It contains both an x^3 term and an x^2 term, indicating that the division produced a cubic leading term alongside a quadratic term. The presence of 3x^3 suggests the dividend had an x^4-type contribution or a combination that, after division, yields an x^3 term, while −5x^2 shows there is a quadratic term with a negative coefficient in the quotient. The key idea is to track how each term in the dividend, when divided by the divisor, contributes to corresponding powers of x in the quotient, and to ensure consistent algebraic combination of like terms. The mismatch with the other options (missing the x^3 term or misplacing signs) confirms why this is the correct structure.

Option 4: 3x^3 + 35. This option includes the correct x^3 term but replaces the quadratic term with a constant 35, which would require a very specific cancellation pattern or a different dividend/divisor arrangement. In standard polynomial division, a constant term in the quotient would come from a constant term in the remainder after division, not as a direct additive term in the quotient if we started from a polynomial with x-dependent terms. The appearance of a constant 35 here does not align with the given dividend structure and the expected x-dependent terms, making this option inconsistent with the anticipated outcome.

Summary and takeaway:
- The presence or absence of x^3 terms is a crucial indicator of the correct quotient degree in polynomial division. 
- The correct option (3x^3 − 5x^2) matches the expected combination of highest-degree and next-highest-degree terms arising from dividing the given expression, while the other options fail to reproduce the necessary degree pattern or coefficients.
- Understanding why the x^3 term appears (and why certain lower-degree terms appear with specific signs) helps distinguish the correct quotient from the distractors.

Divide: 3 𝑥 4 + 10 𝑥 3 − 25 𝑥 2 𝑥 + 5

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