101
While playing in a garden, Samaira saw a honeycomb and asked her mother what is that. Her mother replied that it's a honeycomb made by honey bees to store honey. Also, she told her that the shape of the honeycomb formed is a mathematical structure. The mathematical representation of the honeycomb is shown in the graph.
Based on the above information, answer the following questions :
(i) How many zeroes are there for the polynomial represented by the graph given ?
(ii) Write the zeroes of the polynomial.
(iii) (a) If the zeroes of a polynomial $x^2 + (a + 1) x + b$ are $2$ and $-3$, then determine the values of $a$ and $b$.
OR
(iii) (b) If the square of difference of the zeroes of the polynomial $x^2 + px + 45$ is $144$, then find the value of $p$.
Based on the above information, answer the following questions :
(i) How many zeroes are there for the polynomial represented by the graph given ?
(ii) Write the zeroes of the polynomial.
(iii) (a) If the zeroes of a polynomial $x^2 + (a + 1) x + b$ are $2$ and $-3$, then determine the values of $a$ and $b$.
OR
(iii) (b) If the square of difference of the zeroes of the polynomial $x^2 + px + 45$ is $144$, then find the value of $p$.
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(i) Two
(ii) $7$ and $-7$
(iii) (a) $-(a + 1) = 2 + (-3) \Rightarrow a = 0$
$b = 2 \times (-3) \Rightarrow b = -6$
OR
(b) Let $\alpha$ and $\beta$ be the zeroes of given polynomial
Here, $\alpha + \beta = -p$ and $\alpha \beta = 45$
$(\alpha - \beta)^2 = 144$
$\Rightarrow (\alpha + \beta)^2 - 4\alpha\beta = 144$
$\Rightarrow (-p)^2 - 4 \times 45 = 144$
$\Rightarrow p = \pm 18$
(ii) $7$ and $-7$
(iii) (a) $-(a + 1) = 2 + (-3) \Rightarrow a = 0$
$b = 2 \times (-3) \Rightarrow b = -6$
OR
(b) Let $\alpha$ and $\beta$ be the zeroes of given polynomial
Here, $\alpha + \beta = -p$ and $\alpha \beta = 45$
$(\alpha - \beta)^2 = 144$
$\Rightarrow (\alpha + \beta)^2 - 4\alpha\beta = 144$
$\Rightarrow (-p)^2 - 4 \times 45 = 144$
$\Rightarrow p = \pm 18$