91Ó°ÊÓ

Factoring quadratics is a fundamental algebraic skill that involves breaking down a quadratic expression into simpler, multiplicative components called factors. Let's explore the method for factoring the expression given in the problem, which is \(x^2 + 4x - 21\).

For quadratic expressions in the form \(ax^2 + bx + c\), we start by identifying the coefficients \(a = 1\), \(b = 4\), and \(c = -21\).

• The next step is to find two numbers that multiply to \(-21\) (since \(a \times c = 1 \times -21\)) and add up to \(4\), the coefficient of the linear term.
• In this example, the numbers \(7\) and \(-3\) are the right pair: \(7 \times -3 = -21\) and \(7 + (-3) = 4\).

With these numbers found, we then rewrite the middle term as \(x^2 + 7x - 3x - 21\) and factor by grouping:

• Group the terms to have: \((x^2 + 7x) + (-3x - 21)\).
• Factor out the greatest common factors: \(x(x + 7) - 3(x + 7)\).
• Finally, factor by grouping to get the expression fully factored: \((x - 3)(x + 7)\).

This process transforms the quadratic into a product of two binomials, allowing us to solve equations easier when set to zero.

Solving quadratic equations is the process of finding the values for the variable that make the equation true. The given equation is \(x^2 + 4x - 21 = 0\). To solve it, we will use the factored form from the previous section, \((x - 3)(x + 7)\).

The principle is simple: If a product of two factors equals zero, then at least one of the factors must be zero. This is known as the zero-product property, and it provides us with two smaller equations to solve:

• Set \(x - 3 = 0\): Solve for \(x\) by adding \(3\) to both sides giving \(x = 3\).
• Set \(x + 7 = 0\): Solve for \(x\) by subtracting \(7\) from both sides providing \(x = -7\).

These two solutions, \(x = 3\) and \(x = -7\), are the points where the original quadratic equation holds true. Both solutions are significant because they indicate where the graph of the quadratic touches the x-axis when plotted. These points of intersection are called the roots of the equation.

Polynomial factoring is a broader concept that forms the backbone of algebraic manipulation and involves expressing a polynomial as a product of simpler, non-factorizable polynomials. In the realm of quadratics, such as \(x^2 + 4x - 21\), it specifically involves breaking it down into binomials, as we've seen in this exercise.

Polynomials can have varying degrees, and each degree can affect how you approach factoring:

• For quadratics (degree 2), the standard factorization method includes finding two numbers that fulfill both the multiplication and addition conditions for the coefficients \(a, b, \text{ and } c\).
• Higher-degree polynomials might require more advanced techniques, including synthetic division, long division, or using the rational root theorem to find factors.

Factoring is essential not only for solving equations but also for simplifying expressions, finding zeros of polynomial functions, and in calculus for integration and differentiation purposes. Understanding how to factor efficiently allows deeper insight into the behavior of polynomial functions and their graphs.