Question

1) Draw the graph for the polynomial
P(x) = x² - 5x+6 and find the zeros
from the graph.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomial is

$\rm :\longmapsto\:p(x) = {x}^{2} - 5x + 6$

$\rm :\longmapsto\:Let \: y \: = {x}^{2} - 5x + 6$

Substituting 'x = 0' in the given equation, we get

$\rm :\longmapsto\: y \: = {0}^{2} - 5 \times 0+ 6$

$\rm :\longmapsto\: y \: = 6$

Substituting 'x = 1' in the given equation, we get

$\rm :\longmapsto\: y \: = {1}^{2} - 5 \times 1 \: + \: 6$

$\rm :\longmapsto\:y = 1 - 5 + 6$

$\rm :\longmapsto\: y \: = 2$

Substituting 'x = 2' in the given equation, we get

$\rm :\longmapsto\: y \: = {2}^{2} - 5 \times 2+ 6$

$\rm :\longmapsto\:y = 4 - 10 + 6$

$\rm :\longmapsto\:y = 10 - 10$

$\rm :\longmapsto\:y = 0$

Substituting 'x = - 1' in the given equation, we get

$\rm :\longmapsto\: y \: = { (- 1)}^{2} - 5 \times ( - 1)+ 6$

$\rm :\longmapsto\:y = 1 + 5 + 6$

$\rm :\longmapsto\:y = 12$

Substituting 'x = 3' in the given equation, we get

$\rm :\longmapsto\: y \: = {3}^{2} - 5 \times 3+ 6$

$\rm :\longmapsto\:y = 9 - 15 + 6$

$\rm :\longmapsto\:y = 15- 15$

$\rm :\longmapsto\:y = 0$

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 6 \\ \\ \sf 1 & \sf 2\\ \\ \sf 2 & \sf 0\\ \\ \sf - 1 & \sf 12\\ \\ \sf 3 & \sf 0 \end{array}} \\ \end{gathered}$

We know,

The graph of the polynomial p(x) = ax² + bx + c is in the shape of parabola whose vertex is

$\rm :\longmapsto\: \bigg(-\dfrac{b}{2a},\dfrac{4ac - {b}^{2} }{4a}\bigg)$

So, vertex of p(x) = x² - 5x + 6 is

$\rm : = \: \bigg(-\dfrac{(- 5)}{2 \times 1},\dfrac{4 \times 1 \times 6 - {5}^{2} }{4 \times 1}\bigg)$

$\rm : = \: \bigg(\dfrac{5}{2},\dfrac{24- 25}{4}\bigg)$

$\rm : = \: \bigg(\dfrac{5}{2}, - \: \dfrac{1}{4}\bigg)$

$\rm : = \: \bigg(2.5, \: - \: 0.25\bigg)$

➢ Now draw a graph using the points.

➢ See the attachment graph.

Zero of the polynomial p(x) means those values of x for which p(x) =y = 0

Hence,

From graph, the zeroes of p(x) are 2 and 3.