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: Understanding the Question:
The question asks for the number of zeroes of the given polynomial \(p(x) = (x - 2)(x + 3)\).
A zero of a polynomial is a value of the variable \(x\) that makes the polynomial equal to zero.

Step 2: Key Formula or Approach:
The degree of a polynomial determines the maximum number of real zeroes it can have.
For a polynomial expressed in factored form, the zeroes can be directly found by setting each factor to zero.
Alternatively, expanding the polynomial shows its degree:
\[ p(x) = (x - 2)(x + 3) = x^2 + x - 6 \]
This is a quadratic polynomial, which has a degree of 2.

Step 3: Detailed Explanation:
1. Given polynomial: \(p(x) = (x - 2)(x + 3)\).
2. To find the zeroes of \(p(x)\), we set the polynomial equal to zero:
\[ p(x) = 0 \]
\[ (x - 2)(x + 3) = 0 \]
3. By the zero-product property, if the product of two factors is zero, at least one of the factors must be zero.
4. Therefore, either:
\[ x - 2 = 0 \implies x = 2 \]
or
\[ x + 3 = 0 \implies x = -3 \]
5. This gives us two distinct values of \(x\) that satisfy the equation: \(x = 2\) and \(x = -3\).
6. Thus, the polynomial has exactly two zeroes.

Step 4: Final Answer:
Hence, the correct option is (C).