Question

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ a_i, b_i \in \mathbb{R} $ for $ i \in \{1, 2, 3\} $.
Define the functions $ f: \mathbb{R} \to \mathbb{R},\ g: \mathbb{R} \to \mathbb{R},\ h: \mathbb{R} \to \mathbb{R} $ by:
$$ f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\quad g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, $$ $$ h(x) = f(x+1) - g(x+2) $$ If $ f(x) \ne g(x) $ for every $ x \in \mathbb{R} $, then the coefficient of $ x^3 $ in $ h(x) $ is:

• A. 8
• B. 2
• C. -4
• D. -6

Hint:

When subtracting polynomials after shifting input values, expand only relevant terms to find specific coefficients efficiently.

Answer 1

The Correct Option is C

To find the coefficient of \( x^3 \) in the function \( h(x) = f(x+1) - g(x+2) \), we first need to expand each component.

Step 1: Expand \( f(x+1) \)

Starting with \( f(x+1) = a_1 + 10(x+1) + a_2(x+1)^2 + a_3(x+1)^3 + (x+1)^4 \), expand each term:

1. \( 10(x+1) = 10x + 10 \)

2. \( a_2(x+1)^2 = a_2(x^2 + 2x + 1) = a_2x^2 + 2a_2x + a_2 \)

3. \( a_3(x+1)^3 = a_3(x^3 + 3x^2 + 3x + 1) = a_3x^3 + 3a_3x^2 + 3a_3x + a_3 \)

4. \( (x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1 \)

Collecting terms for the \( x^3 \) coefficient in \( f(x+1) \):

\( a_3x^3 + 4x^3 = (a_3 + 4)x^3 \)

Step 2: Expand \( g(x+2) \)

Next, expand \( g(x+2) = b_1 + 3(x+2) + b_2(x+2)^2 + b_3(x+2)^3 + (x+2)^4 \):

1. \( 3(x+2) = 3x + 6 \)

2. \( b_2(x+2)^2 = b_2(x^2 + 4x + 4) = b_2x^2 + 4b_2x + 4b_2 \)

3. \( b_3(x+2)^3 = b_3(x^3 + 6x^2 + 12x + 8) = b_3x^3 + 6b_3x^2 + 12b_3x + 8b_3 \)

4. \( (x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16 \)

Collecting terms for the \( x^3 \) coefficient in \( g(x+2) \):

\( b_3x^3 + 8x^3 = (b_3 + 8)x^3 \)

Step 3: Calculate the coefficient of \( x^3 \) in \( h(x) \)

The coefficient of \( x^3 \) in \( h(x) = f(x+1) - g(x+2) \) is:

\((a_3 + 4) - (b_3 + 8) = a_3 + 4 - b_3 - 8 = a_3 - b_3 - 4\)

Step 4: Conclusion

Since \( f(x) \neq g(x) \) for any \( x \), it implies \( a_3 \neq b_3 \), the specific values don't make a difference to the meaning of this expression, the coefficient remains constant as:-4