Question

If one root is reciprocal of the other root x^2 - 4x + k, then find the value of k. ​

Answer 1

$\textsf{\large{\underline{Solution}:}}$

Given:

→ f(x) = x² - 4x + k

Comparing f(x) with ax² + bx + c, we get:

$\sf\implies \begin{cases}\sf a=1\\ \sf b=-4\\ \sf c=k\end{cases}$

> Let α and β the zeros of f(x).

We know that:

$\sf\implies Product\ of\ Zeros=\dfrac{c}{a}$

Therefore:

$\sf\implies \alpha\beta=\dfrac{c}{a}$

$\sf\implies \alpha\beta=\dfrac{k}{1}$

$\sf\implies \alpha\beta=k$

As one root is the reciprocal of other, their product must be 1. Therefore:

$\sf\implies k=1\ \ \ (Answer)$

$\textsf{\large{\underline{Answer}:}}$

• So, the value of k is 1.
• Therefore the polynomial is - x² - 4x + 1

$\textsf{\large{\underline{Know More}:}}$

1. Relationship between zeros and coefficients (Quadratic Polynomial)

Let f(x) = ax² + bx + c and let α and β be the zeros of f(x).

Therefore:

$\sf\implies\alpha+\beta=\dfrac{-b}{a}$

$\sf\implies\alpha\beta=\dfrac{c}{a}$

2. Relationship between zeros and coefficients (Cubic Polynomial)

Let f(x) = ax³ + bx² + cx + d and let α, β and γ be the zeros of f(x).

Therefore:

$\sf\implies \alpha+\beta+\gamma=\dfrac{-b}{a}$

$\sf\implies \alpha\beta+\beta\gamma+\alpha\gamma=\dfrac{c}{a}$

$\sf\implies \alpha\beta\gamma=\dfrac{-d}{a}$

Answer 2

Answer:

=> f (x) = x^2 - 4x + k [ given ]

=> a = 1, b = - 4, c = k

=> Let a and ß be the zeros of f (x)

=> Product of zeros = c / a

=> a ß = c / a

=> a ß = k / 1

=> a ß = k

=> Therefore, k = 1

Step-by-step explanation:

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