Find the zeroes of the polynomial p(x) = $x^2 - 5x + 6$ and verify the relationship between the zeroes and the coefficients.

Step 1: Find the zeroes by factorization

To find the zeroes of the polynomial, we set $p(x) = 0$. We then factor the quadratic expression $x^2 - 5x + 6$. We look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. We split the middle term, $-5x$, into $-2x - 3x$ and then factor by grouping.

Step 2: Determine the zeroes

From the factored form $(x - 2)(x - 3) = 0$, we use the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us the two zeroes of the polynomial.

Step 3: Identify coefficients

For a quadratic polynomial in the standard form $ax^2 + bx + c$, we identify the coefficients $a$, $b$, and $c$. In our polynomial $x^2 - 5x + 6$, the coefficient of $x^2$ is $a=1$, the coefficient of $x$ is $b=-5$, and the constant term is $c=6$.

Step 4: Verify sum of zeroes

Let the zeroes be $\alpha = 2$ and $\beta = 3$. The sum of the zeroes is $\alpha + \beta = 2 + 3 = 5$. According to the relationship between zeroes and coefficients, the sum of the zeroes is also equal to $-b/a$. We calculate $-b/a = -(-5)/1 = 5$. Since both values are equal, the relationship is verified.

Step 5: Verify product of zeroes

The product of the zeroes is $\alpha \beta = 2 \times 3 = 6$. According to the relationship between zeroes and coefficients, the product of the zeroes is also equal to $c/a$. We calculate $c/a = 6/1 = 6$. Since both values are equal, the relationship is verified.

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