In an algebraic simplification question, let x be a real number such that x − 1/x = 3. Using this relation, evaluate the expression (2x^4 + 3x^3 + 13x^2 − 3x + 2) ÷ (3x^4 + 3) and find its exact numerical value.

Introduction / Context:
This aptitude question checks how well you can use an algebraic condition to simplify a complicated rational expression in terms of x. Instead of solving for x directly and getting messy values, we use the relation x − 1/x = 3 to reduce higher powers of x to something simple and then evaluate the full expression. This type of question is common in competitive exams to test algebraic manipulation, identities, and simplification skills.

Given Data / Assumptions:

• x is a real number and x ≠ 0 because 1/x is defined.
• The relation x − 1/x = 3 holds.
• We need the value of (2x^4 + 3x^3 + 13x^2 − 3x + 2) ÷ (3x^4 + 3).
• No approximate decimal value is required, only an exact simplified fraction.

Concept / Approach:
The key idea is to use the relation x − 1/x = 3 to form an equation in x and then reduce higher powers of x. From x − 1/x = 3, we obtain a quadratic equation in x. Once we know that x satisfies this quadratic, we can replace x^2 or x^4 in the given expression using that relation. Another approach is to solve the quadratic for x and then substitute each root into the expression. Since both roots satisfy the same equation, the value of the expression will be the same for both roots, giving us a unique answer.

Step-by-Step Solution:
Step 1: Start from x − 1/x = 3.Step 2: Multiply both sides by x to remove the denominator: x^2 − 1 = 3x.Step 3: Rearrange to get a quadratic: x^2 − 3x − 1 = 0.Step 4: This shows x satisfies x^2 = 3x + 1. Hence x^4 = (x^2)^2 = (3x + 1)^2 = 9x^2 + 6x + 1.Step 5: Substitute x^4 into the numerator: 2x^4 + 3x^3 + 13x^2 − 3x + 2 = 2(9x^2 + 6x + 1) + 3x^3 + 13x^2 − 3x + 2.Step 6: Simplify the numerator: 18x^2 + 12x + 2 + 3x^3 + 13x^2 − 3x + 2 = 3x^3 + 31x^2 + 9x + 4.Step 7: Similarly, substitute x^4 into the denominator: 3x^4 + 3 = 3(9x^2 + 6x + 1) + 3 = 27x^2 + 18x + 6.Step 8: Now the expression is (3x^3 + 31x^2 + 9x + 4) ÷ (27x^2 + 18x + 6).Step 9: Instead of further algebraic reduction, use the quadratic x^2 − 3x − 1 = 0 to find numerical values for x and substitute.Step 10: The quadratic has two real roots. Substituting either root into the simplified fraction gives the same constant value, which is 4/3.

Verification / Alternative check:
An efficient verification method is to solve x^2 − 3x − 1 = 0 using the quadratic formula, compute x approximately, and then evaluate the original fraction numerically with a calculator. Doing this for both roots shows that the fraction is approximately 1.3333 in each case, which corresponds to 4/3 when written as an exact fraction. Since the value is independent of the chosen root, the simplified result is confirmed as 4/3.

Why Other Options Are Wrong:
Options 1/3 and 2/3 are much smaller than the evaluated expression, and substitution of any root yields a value significantly larger than these. Option 5/3 is approximately 1.6667, which does not match the actual value around 1.3333. The option 7/3 is even larger and clearly inconsistent. Only 4/3 matches the exact algebraic simplification and numerical verification.

Common Pitfalls:
Many learners try to expand and simplify the large polynomial expression directly without using the relation x − 1/x = 3, which leads to lengthy calculations and mistakes. Another common error is to assume that x has a single value and not check that the expression remains constant for all possible roots. Some students also square the relation incorrectly or mishandle denominators. Using the quadratic relation carefully and checking work avoids these issues.

Final Answer:
The exact value of the expression (2x^4 + 3x^3 + 13x^2 − 3x + 2) ÷ (3x^4 + 3) under the condition x − 1/x = 3 is 4/3.