Question

If one root of the polynomial 5x3 + 13x + k is reciprocal of the other, then find the value of k?

Answer 1

Corrected Question:

If one root of the polynomial 5x² + 13x + k is reciprocal of the other, then find the value of k?

Solution:

As per the given question, we have:

• Quadratic polynomial = 5x²+13x+k
• One root of polynomial is reciprocal of other.

Let us suppose the roots of the given polynomial as α and 1/α.

We know, Product of zeroes is equal to the constant term by coefficient of x². So,

$\longmapsto\sf{Product\: of\: zeroes=\dfrac{Constant\: term}{Coefficient\: of\: {x}^{2}}}$

Substituting, constant term as k and coefficient of x² as 5.

$\longmapsto\sf{Product\: of\: zeroes=\dfrac{k}{5}}$

Now, Let's take product of zeroes as α and 1/α,

$\longmapsto\sf{Product\: of\: zeroes=\alpha\times \dfrac{1}{\alpha}}$

Multiplying, α and 1/α,

$\longmapsto\sf{Product\: of\: zeroes=1}$

As, the product of zeroes also equals to k/5. So,

$\longmapsto\sf{\dfrac{k}{5}=1}$

Transposing, 5 from LHS to RHS,

$\longmapsto\sf{k=1\times 5}$

$\longmapsto\bold{k=5}$

Required Answer:

$\: \: \: \: \: \leadsto$ Henceforth, value of k is 5.