If the polynomial (2x3 + ax2 + 3x − 5) and (x3 + x2 − 2x + a) leave the same remainder when divided by (x−2), find the remainder.
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Correct answer: D
Explanation
The correct answer is option 4 i.e. 5.
Since both polynomials leave the same remainder when divided by (x-2), we can equate the remainders:
P1(2) = P2(2)
where P1(x) = 2x³ + ax² + 3x - 5 and P2(x) = x³ + x² - 2x + a.
So, we have:
⇒ 2(2)³ + a(2)² + 3(2) - 5 = 2³ + 2² - 2(2) + a
⇒ 16 + 4a + 6 - 5 = 8 + 4 - 4 + a
Simplifying, we get:
⇒ 4a + 17 = a + 8
⇒ 3a = -9
⇒ a = -3
Now that we know a = -3, we can substitute it into either P1(x) or P2(x) and evaluate it at x = 2 to find the remainder.
Let's use P1(x):
⇒ P1(2) = 2(2)³ + (-3)(2)² + 3(2) - 5 = 16 - 12 + 6 - 5 = 5
Therefore, the remainder when both polynomials are divided by (x-2) is 5.