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Let p(x) be a quadratic polynomial with constant term 1. Suppose p(x) when divided by x – 1 leaves remainder 2 and when divided by x + 1 leaves remainder 4. Then the sum of the roots of p(x) = 0 is
  • a)
    –1
  • b)
    1
  • c)
    -1/2
  • d)
    1/2
Correct answer is option 'D'. Can you explain this answer?
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Let p(x) be a quadratic polynomial with constant term 1. Suppose p(x) ...
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Let p(x) be a quadratic polynomial with constant term 1. Suppose p(x) ...
Understanding the Problem
We are given a quadratic polynomial p(x) of the form:
- p(x) = ax^2 + bx + 1
where the constant term is 1.
Given Conditions
1. Remainder when p(x) is divided by x - 1 is 2:
- p(1) = 2
2. Remainder when p(x) is divided by x + 1 is 4:
- p(-1) = 4
Setting Up the Equations
Using the given conditions, we can set up the following equations:
- From p(1) = 2:
- a(1)^2 + b(1) + 1 = 2
- This simplifies to: a + b + 1 = 2
- Therefore, a + b = 1 (Equation 1)
- From p(-1) = 4:
- a(-1)^2 + b(-1) + 1 = 4
- This simplifies to: a - b + 1 = 4
- Therefore, a - b = 3 (Equation 2)
Solving the Equations
Now, we solve these two equations:
1. a + b = 1
2. a - b = 3
Adding both equations, we get:
- 2a = 4
- Hence, a = 2
Substituting a back into Equation 1:
- 2 + b = 1
- Thus, b = -1
Finding the Roots
Now, substituting a and b into the polynomial:
- p(x) = 2x^2 - x + 1
The sum of the roots of a quadratic polynomial p(x) = ax^2 + bx + c is given by:
- Sum of roots = -b/a
Substituting our values:
- Sum of roots = -(-1)/2 = 1/2
Conclusion
Thus, the sum of the roots of p(x) = 0 is:
- 1/2, which corresponds to option 'D'.
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Let p(x) be a quadratic polynomial with constant term 1. Suppose p(x) when divided by x – 1 leaves remainder 2 and when divided by x + 1 leaves remainder 4. Then the sum of the roots of p(x) = 0 isa)–1b)1c)-1/2d)1/2Correct answer is option 'D'. Can you explain this answer?
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