Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(A) $p(0) = 5$.

(B) The polynomial $p(x) - q(x)$ cuts the $x$-axis at $(3, 0)$.

(C) The sum $p(x) + q(x)$ is equal to $6x + 4$ for all real $x$.

Find the polynomials $p(x)$ and $q(x)$.

From (A): $p(0) = b = 5$.

From (B): $p(x) + q(x) = (a + c)x + (b + d)$

$= 6x + 4$

$\Rightarrow a + c = 6$ and $b + d = 4$.

Since $b = 5$, $d = 4 - 5 = -1$.

From (B): $p(x) - q(x) = (a - c)x + (b - d)$.

At $x = 3$, value = 0:

$3(a - c) + (5 - (-1)) = 0$

$\Rightarrow 3(a - c) + 6 = 0$

$\Rightarrow a - c = -2$.

Solve $a + c = 6$ and $a - c = -2$: $a = 2$, $c = 4$.

Therefore, $p(x) = 2x + 5$ and $q(x) = 4x - 1$.