Question
Download Solution PDFIf the polynomial (2x3 + ax2 + 3x - 5) and (x3 + x2 - 2x + a) leave the same remainder when divided by (x-2), find the remainder.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
If the polynomial (2x3 + ax2 + 3x - 5) and (x3 + x2 - 2x + a) leave the same remainder when divided by (x-2), find the remainder.
Formula used:
For a polynomial f(x), the remainder when divided by (x-c) is f(c).
Calculation:
Let f(x) = 2x3 + ax2 + 3x - 5 and g(x) = x3 + x2 - 2x + a
We need to find f(2) and g(2) and set them equal:
f(2) = 2(2)3 + a(2)2 + 3(2) - 5
⇒ f(2) = 2(8) + a(4) + 6 - 5
⇒ f(2) = 16 + 4a + 6 - 5
⇒ f(2) = 17 + 4a
g(2) = (2)3 + (2)2 - 2(2) + a
⇒ g(2) = 8 + 4 - 4 + a
⇒ g(2) = 8 + a
Since f(2) = g(2), we have:
17 + 4a = 8 + a
⇒ 17 + 4a - a = 8
⇒ 3a = 8 - 17
⇒ 3a = -9
⇒ a = -3
Now, we find the remainder when the polynomials are divided by (x-2) using the value of a:
f(2) = 17 + 4(-3)
⇒ f(2) = 17 - 12
⇒ f(2) = 5
∴ The correct answer is option (4).
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