### Theory:

If the value of the polynomial $$p(x)$$ at $$x = k$$ is zero $$p(k) = 0$$, then the real number $$k$$ is called the zero of the polynomial $$p(x)$$.
Important!
To find the zeroes of a quadratic polynomial, put $$p(x) = 0$$.
Example:
1. Find the zeroes of the quadratic expression $$p(x) = 2x^2 + 3x + 1$$.

Solution:

$$p(x) = 2x^2 + 3x + 1$$

Put $$p(x) = 0$$.

$$2x^2 + 3x + 1 = 0$$

$$2x^2 + 2x + x + 1 = 0$$

$$2x(x + 1) + 1(x + 1) = 0$$

$$(2x + 1) (x + 1) = 0$$

$$2x + 1 = 0$$ or $$x + 1 = 0$$

$$x = \frac{-1}{2}$$ or $$x = -1$$

Thus, the zeroes of $$p(x) = 2x^2 + 3x + 1$$ are $$\frac{-1}{2}$$ and $$-1$$.

2. Check $$2$$ is a zero of the polynomial $$p(x) = 3x^2 - 5x - 2$$.

Solution:

$$p(x) = 3x^2 - 5x - 2$$

Put $$x = 2$$ in $$p(x)$$.

$$p(2) = 3(2)^2 - 5(2) - 2$$

$$= 3(4) - 10 - 2$$

$$= 12 - 12$$

$$= 0$$

Thus, $$2$$ is a zero of the polynomial $$p(x) = 3x^2 - 5x - 2$$.