If one root of the polynomial f(x) = 5x² + 13x + k is the reciprocal of the other, find the value of k

Video Explanation

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Given

f(x) = 5x² + 13x + k

Let the roots of the polynomial be α and β.

It is given that one root is the reciprocal of the other.

To Find

The value of k.

Solution

If one root is the reciprocal of the other, then:

αβ = 1

For a quadratic polynomial ax² + bx + c:

αβ = c/a

Step 1: Compare Coefficients

Comparing f(x) = 5x² + 13x + k with ax² + bx + c,

a = 5,   b = 13,   c = k

Step 2: Use the Condition αβ = 1

αβ = c/a

⇒ k/5 = 1

⇒ k = 5

Final Answer

k = 5

Conclusion

Hence, if one root of the polynomial f(x) = 5x² + 13x + k is the reciprocal of the other, then the value of k is 5.