What are the factors of $x^2-5 x+6$ ?
To factor the quadratic polynomial \(x^2 - 5x + 6\), we look for two factors of the form \((x - r_1)\) and \((x - r_2)\). When we expand the product \((x - r_1)(x - r_2)\), we get:
\[(x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + (r_1 \times r_2).\]
Comparing this to our given expression \(x^2 - 5x + 6\), we see that the sum of the roots \(r_1 + r_2\) must equal \(5\), and the product of the roots \(r_1 \times r_2\) must equal \(6\).
We need to find two numbers that multiply to \(6\) and add up to \(5\). Let's list the factor pairs of \(6\):
The pair \((2, 3)\) satisfies both conditions. Therefore, we can write the polynomial as:
\[x^2 - 5x + 6 = (x - 2)(x - 3).\]
Comparing this result to the given options, we find that it matches option B.
Based on the factorization, the factors of \(x^2 - 5x + 6\) are \((x - 2)\) and \((x - 3)\). This corresponds to option B. The final answer is \(\boxed{B}\).
Quadratic factoring is essential in physics and engineering, such as determining the time when a thrown ball reaches a certain height. If the height \(h\) of an object at time \(t\) is given by a quadratic equation like \(h(t) = -5t^2 + 25t + 30\), factoring helps identify the roots (the specific times when the object is at a certain height or hits the ground). By setting the expression to zero, we can factor out common terms to easily solve for the time \(t\). This application demonstrates how algebraic techniques simplify complex physical motion problems into manageable calculations.