91Ó°ÊÓ

A quadratic expression is a type of polynomial that you might frequently encounter in algebra. It has the general form of \(ax^2 + bx + c\), where:

• \(a\), \(b\), and \(c\) are constants
• \(x\) is the variable
• \(a eq 0\) to ensure that the expression is quadratic

The expression \(x^2 - 14x + 48\) has coefficients where \(a = 1\), \(b = -14\), and \(c = 48\). Quadratic expressions are distinct due to their characteristic "squared" term, which provides the U-shaped graph known as a parabola when plotted on a coordinate plane. Understanding the composition of a quadratic expression is the first step towards mastering techniques like completing the square or factoring.

Rewriting a quadratic expression in its factored form is key to simplifying, solving, or graphing the expression. The factored form of a quadratic expression \(ax^2 + bx + c\) is typically \((px + q)(rx + s)\), where multiplying out the factors will give back the original quadratic expression.
In factored form:

• \(p\) and \(r\) are the factors of \(a\)
• \(q\) and \(s\) are specific numbers that replicate the relationships set by \(b\) and \(c\)

For our example, \(x^2 - 14x + 48\), after examining potential factor combinations, we found that the expression can be broken down into \((x - 6)(x - 8)\). Each binomial represents a solution to the quadratic equation when set to zero, showing the roots of the equation. Recognizing the factored form allows you to quickly determine where the graph of the expression crosses the x-axis.

Polynomial factoring is a method used to simplify polynomials like quadratic expressions by expressing them as a product of smaller degree polynomials, ideally linear ones. This process can help in solving equations, simplifying expressions, and even in applications like calculating certain limits in calculus.
The key steps in the process involve:

• Identifying terms \(b\) and \(c\) in the quadratic expression \(ax^2 + bx + c\)
• Searching for two numbers that multiply to \(c\) and add to \(b\)
• Rewriting the quadratic into a product of two binomials

In our task, after analyzing the polynomial \(x^2 - 14x + 48\), we found that \(-6\) and \(-8\) fulfill both conditions: their product gives \(48\) and their sum provides \(-14\). Factoring polynomials like these is valuable for finding solutions to equations and plays a crucial role in more advanced mathematical operations.