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Let a and b be two distinct roots of a p...

Let a and b be two distinct roots of a polynomial equation f (x) = 0, Then there exists at least one root lying between a and b of polynomial equation.

A

a) f(x) =0

B

b) f'(x) 0

C

c) f''(x) =0

D

d) None of these

Text Solution

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The correct Answer is:
To solve the problem, we will use the Intermediate Value Theorem, which is a fundamental theorem in calculus. The theorem states that if a function is continuous on a closed interval [a, b] and takes on different values at the endpoints, then it must take on every value between those endpoints at least once. ### Step-by-Step Solution: 1. **Identify the Function and Roots**: Let \( f(x) \) be a polynomial function such that \( f(a) = 0 \) and \( f(b) = 0 \), where \( a \) and \( b \) are two distinct roots of the polynomial. 2. **Check Continuity**: Since \( f(x) \) is a polynomial, it is continuous everywhere on the real line, including the interval \([a, b]\). 3. **Evaluate the Function at the Endpoints**: We know that: - \( f(a) = 0 \) - \( f(b) = 0 \) 4. **Apply the Intermediate Value Theorem**: According to the Intermediate Value Theorem, since \( f(x) \) is continuous on \([a, b]\) and \( f(a) \) and \( f(b) \) are both equal to 0, there must be at least one point \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \). 5. **Conclusion**: Therefore, we conclude that there exists at least one root of the polynomial \( f(x) = 0 \) in the interval \((a, b)\).
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Knowledge Check

  • Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

    A
    f(x) =0
    B
    f'(x)=0
    C
    f''(x)=0
    D
    none of these

  • The number of real roots of the polynomial equation x^(4)-x^(2)+2x-1=0 is

    A
    0
    B
    2
    C
    3
    D
    4

  • The root of the equation f (x) = 0 in the interval (a,b) is given by

    A
    `(af(b) - bf(a))/(b - a)`
    B
    `(b f(a) - a f(b))/(f(b) - f(a))`
    C
    `(af(b) - bf(a))/(f(b) - f(a))`
    D
    none of these

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