The graph of a linear polynomial $p(x)$ passes through the points $(1, 5)$ and $(3, 11)$.

(A) Find the polynomial $p(x)$.
(B) Find the coordinates where the graph of $p(x)$ cuts the axes.
(C) Draw the graph of $p(x)$ and verify your answers.

(A) Let $p(x) = ax + b$

At (1, 5): $a + b = 5$ ...(i)

At (3, 11): $3a + b = 11$ ...(ii)

Subtract (i) from (ii):

$2a = 6 \Rightarrow a = 3$; then $b = 2$.

So, $p(x) = 3x + 2$.

(B) Cuts $y$-axis ($x = 0$): $y = p(0) = 2$, i.e., $(0, 2)$.

Cuts $x$-axis ($y = 0$):

$3x + 2 = 0 \Rightarrow x = -\frac{2}{3}$, i.e., $\left(-\frac{2}{3}, 0\right)$.

(C) From (B), by the points $(0, 2)$ and $(-\frac{2}{3}, 0)$ at the $y$-axis and $x$-axis respectively, the straight line passes through them.

Concept Applied
A linear polynomial has the form $ax + b$ ($a
eq 0$), and its graph is always a straight line.