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x^50+2x^37+p is divisible by (x+1) then ...

`x^50+2x^37+p` is divisible by (x+1) then the value of p is:

A

-3

B

-2

C

1

D

2

Text Solution

AI Generated Solution

The correct Answer is:
To find the value of \( p \) such that the polynomial \( f(x) = x^{50} + 2x^{37} + p \) is divisible by \( (x + 1) \), we can use the Remainder Theorem. According to this theorem, if a polynomial \( f(x) \) is divisible by \( (x + 1) \), then \( f(-1) = 0 \). ### Step-by-step Solution: 1. **Set up the polynomial**: \[ f(x) = x^{50} + 2x^{37} + p \] 2. **Apply the Remainder Theorem**: Since \( f(x) \) is divisible by \( (x + 1) \), we need to find \( f(-1) \): \[ f(-1) = (-1)^{50} + 2(-1)^{37} + p \] 3. **Evaluate the powers**: - \( (-1)^{50} = 1 \) (since 50 is even) - \( (-1)^{37} = -1 \) (since 37 is odd) Therefore, we can substitute these values into the equation: \[ f(-1) = 1 + 2(-1) + p \] 4. **Simplify the expression**: \[ f(-1) = 1 - 2 + p = -1 + p \] 5. **Set the expression equal to zero**: Since \( f(-1) = 0 \): \[ -1 + p = 0 \] 6. **Solve for \( p \)**: \[ p = 1 \] ### Conclusion: The value of \( p \) is \( 1 \). ---
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