Question

How many zeroes does \(p(x) = (x - 2)(x + 3)\) have ?

• A. Zero
• B. One
• C. Two
• D. Three

Hint:

A polynomial of degree \(n\) can have at most \(n\) real zeroes.
Since this polynomial is of degree 2 (quadratic), it has at most 2 real zeroes.
Since the two factors are distinct linear factors, they correspond to two distinct real zeroes.

Answer 1

The Correct Option is C

Step 1: Expand the polynomial into standard form.
Instead of reading the zeroes straight off the factors, multiply them out into the standard quadratic form $ax^2+bx+c$:
\[ p(x) = (x-2)(x+3) = x^2+3x-2x-6 = x^2+x-6 \]
So $a=1$, $b=1$, $c=-6$.

Step 2: Use the discriminant to count the zeroes.
The discriminant $D=b^2-4ac$ tells us how many real zeroes a quadratic has: two distinct real zeroes if $D>0$.
\[ D = (1)^2-4(1)(-6) = 1+24 = 25 \]
Since $D=25>0$, the polynomial has two distinct real zeroes.

Step 3: Solve using the quadratic formula to confirm the values.
\[ x = \frac{-b\pm\sqrt{D}}{2a} = \frac{-1\pm5}{2} \]
This gives $x=\dfrac{-1+5}{2}=2$ and $x=\dfrac{-1-5}{2}=-3$, matching the original factors.

Final Answer:
Since there are two distinct real zeroes, option (C) is correct.
\[ \boxed{2} \]