When the polynomial 4x^4 + 10x^3 - 20x^2 + 90 is divided by (x + 2), what is the remainder?

Introduction / Context:
This question tests the remainder theorem for polynomials, which is a very common topic in aptitude and algebra exams. Instead of performing full long division, we can quickly find the remainder when a polynomial in x is divided by a linear factor of the form x - a or x + a by evaluating the polynomial at a suitable value of x.

Given Data / Assumptions:

• Polynomial: P(x) = 4x^4 + 10x^3 - 20x^2 + 90
• Divisor: x + 2
• We assume x is a real number and standard polynomial division rules apply.

Concept / Approach:
The remainder theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a). For a divisor of the form (x + 2), we can rewrite it as (x - (-2)), so the remainder will be P(-2). This avoids long division and gives a fast and accurate result.

Step-by-Step Solution:
Step 1: Identify a such that x + 2 = x - (-2). So a = -2.Step 2: Write P(x) = 4x^4 + 10x^3 - 20x^2 + 90.Step 3: Compute P(-2) = 4(-2)^4 + 10(-2)^3 - 20(-2)^2 + 90.Step 4: Evaluate powers: (-2)^4 = 16, (-2)^3 = -8, (-2)^2 = 4.Step 5: Substitute: P(-2) = 4*16 + 10*(-8) - 20*4 + 90.Step 6: Simplify term by term: 4*16 = 64, 10*(-8) = -80, -20*4 = -80.Step 7: Add all terms: 64 - 80 - 80 + 90 = (64 - 80) = -16, then -16 - 80 = -96, and -96 + 90 = -6.Step 8: Therefore, the remainder when P(x) is divided by (x + 2) is -6.

Verification / Alternative check:
We can verify by performing synthetic division with -2 as the root. Using the coefficients 4, 10, -20, 0, 90 and applying synthetic division with -2 will produce a final remainder of -6. This confirms that our direct substitution using the remainder theorem is correct.

Why Other Options Are Wrong:

• 0: This would mean x + 2 is an exact factor of the polynomial, which is not true because P(-2) is -6, not 0.
• 45: This value is not supported by any correct evaluation method for P(-2).
• 90: This is simply the constant term of the polynomial, not the remainder after division by (x + 2).

Common Pitfalls:
Students sometimes plug in x = 2 instead of x = -2 because they forget that x + 2 corresponds to x - (-2). Another common error is incorrect evaluation of powers of negative numbers, especially confusing (-2)^3 with -8 and (-2)^4 with 16. Careless arithmetic in combining the terms can also lead to wrong answers.

Final Answer:
The remainder when 4x^4 + 10x^3 - 20x^2 + 90 is divided by (x + 2) is -6.