Question

Product of the polynomials \((x^3+8),(x-8)\) is denoted by \(p(x)=ax^4+bx^3+cx^2+dx+e\) then \(p(8)=\)

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is A

To solve the problem, we need to evaluate the product of the given polynomials at \( x = 8 \).

1. Understand the Problem:
We are given the product of two polynomials \( (x^3 + 8) \) and \( (x - 8) \), and we are asked to find the value of \( p(8) \), where:
\[ p(x) = (x^3 + 8)(x - 8) \]

2. Simplify the Expression:
We can simplify the product of the polynomials using the formula \( (a^3 + b^3) = (a + b)(a^2 - ab + b^2) \), where \( a = x \) and \( b = 2 \):
\[ x^3 + 8 = (x + 2)(x^2 - 2x + 4) \]

3. Apply the Factorization:
Thus, we have:
\[ (x^3 + 8)(x - 8) = (x + 2)(x^2 - 2x + 4)(x - 8) \] Now, substitute \( x = 8 \) into the expression:
\[ p(8) = (8 + 2)(8^2 - 2(8) + 4)(8 - 8) = 10 \times (64 - 16 + 4) \times 0 = 10 \times 52 \times 0 = 0 \]

Final Answer:
The correct answer is option (A): 0.