91Ó°ÊÓ

A quadratic expression is a polynomial of degree two. This means its highest exponent is two. The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

In our exercise, the expression given is \(p^2 + 11p + 30\). Here, the quadratic term is \(p^2\) (where the coefficient \(a\) is 1), the linear term is \(11p\), and the constant term is \(30\).

Understanding the structure of a quadratic expression helps us in identifying the necessary steps to factor it. Recognizing this structure is the first step in the process of factoring quadratic equations effectively.

The factored form of a quadratic expression is a way of expressing the quadratic equation in terms of two binomials. For example, the quadratic expression \(p^2 + 11p + 30\) can be written as \((p + m)(p + n)\).

Here, \(m\) and \(n\) are numbers that, when multiplied, give the constant term \(c\) (which is 30 in this case) and, when added, give the coefficient \(b\) of the linear term (which is 11 in this case). So, the challenge is finding the correct \(m\) and \(n\).

For our exercise, the numbers \(5\) and \(6\) multiply to 30 and add up to 11. Thus, our factored form is \((p + 5)(p + 6)\). This method highlights the importance of understanding relationships between coefficients and terms in quadratic expressions.

When factoring quadratic expressions, two key operations often used are the product and sum. The product refers to the result of multiplying two numbers, while the sum is the result of adding these same two numbers.

In the context of our exercise, we need to find numbers whose product is the constant term (30) and whose sum is the linear coefficient (11).

To solve \(p^2 + 11p + 30\), we look for two numbers that multiply to 30 and add to 11. The numbers \(5\) and \(6\) fit these conditions because \(5 \times 30 = 30\) and \(5 + 6 = 11\). Once we have these numbers, we can write the quadratic expression in its factored form: \((p + 5)(p + 6)\).

This use of product and sum is a powerful tool for simplifying and solving quadratic equations.