Question

if A and B are the zeros of the polynomial x square - 5 x + 6 then find a polynomial whose zeroes are and 1/b

Answer 1

$\large{\boxed{\underline{\underline{\red{</p><h2>✰<strong><u>QUESTION</u></strong>✰</h2><p>}}}}}$

• If 'a' and 'b' are zeroes of the polynomial $x^2 - 5x + 6$, then find the polynomial whose zeroes are $\frac{1}{a}$ and $\frac{1}{b}$.

✪ANSWER✪

• $\boxed{ x^2 - \frac{5}{6}x + \frac{1}{6}}$ is the required polynomial.

᯽CONCEPTS᯽

1. If $\alpha$ and $\beta$ are zeroes of a quadratic polynomial

$\:\:\:\:\:\:\:ax^2 + bx + c$, then

• $\alpha + \beta = - \frac{b}{a}$

• $\alpha \beta = \frac{c}{a}$

2. If $\alpha$ and $\beta$ are zeroes of a quadratic polynomial, the polynomial is given by

• $x^2 - (\alpha + \beta )x + \alpha \beta$

⍟CALCULATION⍟

Applying the above relations in the given polynomial, we get.

$\:\:\:\:\:\:\: a + b = - \frac{-5}{1}$

$\implies \boxed{ a + b = 5}$

$\:\:\:\:\:\:\: a \times b = \frac{6}{1}$

$\implies \boxed{ ab = 6}$

Now, we will calculate sum of roots and product of roots of the required polynomial.

Sum of roots

$\:\:\:\:\:\:\: \frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}$

$\implies \frac{1}{a} + \frac{1}{b} = \frac{5}{6}$

Sum of roots is $\frac{5}{6}.$

Product of roots

$\:\:\:\:\:\:\: \frac{1}{a}\frac{1}{b} = \frac{1}{ab}$

$\implies \frac{1}{a}\frac{1}{b} = \frac{1}{6}$

Product of roots is $\frac{1}{6}.$

Quadratic Polynomial

The required polynomial is given by

$\to x^2 - (\frac{1}{a} + \frac{1}{b})x + \frac{1}{a}\frac{1}{b}$

$\to x^2 - \frac{5}{6}x + \frac{1}{6}$