If polynomial ax^3+3x^2-13 and 2x^3-5x+a when divided by (x-2) leaves same remainder. Find value of a.

To solve the problem, we need to find the value of \( a \) such that both polynomials \( ax^3 + 3x^2 - 13 \) and \( 2x^3 - 5x + a \) leave the same remainder when divided by \( x - 2 \). According to the Remainder Theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - c \) is simply \( f(c) \). In this case, our \( c \) is 2.

Step 1: Calculate the Remainders for Both Polynomials

We will evaluate both polynomials at \( x = 2 \).

For the First Polynomial:

Substituting \( x = 2 \) into \( ax^3 + 3x^2 - 13 \):

• First, calculate \( 2^3 \): \( 2^3 = 8 \)
• Then, calculate \( 3 \cdot 2^2 \): \( 3 \cdot 4 = 12 \)

The expression becomes:

Remainder = \( a \cdot 8 + 12 - 13 = 8a - 1 \)

For the Second Polynomial:

Now, substitute \( x = 2 \) into \( 2x^3 - 5x + a \):

• Calculate \( 2 \cdot 2^3 \): \( 2 \cdot 8 = 16 \)
• Calculate \( -5 \cdot 2 \): \( -10 \)

The expression becomes:

Remainder = \( 16 - 10 + a = 6 + a \)

Step 2: Set the Remainders Equal

Since both polynomials leave the same remainder when divided by \( x - 2 \), we can set the two expressions we found for the remainders equal to each other:

8a - 1 = 6 + a

Step 3: Solve for \( a \)

Now, we will solve this equation step by step:

• First, rearrange the equation: \( 8a - a = 6 + 1 \)
• This simplifies to: \( 7a = 7 \)
• Now, divide both sides by 7: \( a = 1 \)

Final Result

The value of \( a \) that makes both polynomials leave the same remainder when divided by \( x - 2 \) is \( 1 \).