Lesson 6

Previously, students learned that a quadratic expression in factored form can be quite handy in revealing the zeros of a function and the \(x\)-intercepts of its graph. They also observed that the factored form can also help us solve quadratic equations algebraically. In this lesson, students begin to rewrite quadratic expressions from standard to factored form.

In an earlier unit, students learned to expand quadratic expressions in factored form and rewrite them in standard form. They did so by applying the distributive property to multiply out the factors—first by using diagrams for support, and then by relying on structure they observed in the process. The attention to structure continues in this lesson. Students relate the numbers in the factored form to the coefficients of the terms in standard form, looking for structure that can be used to go in reverse—from standard form to factored form (MP7).

This lesson only looks at expressions of the form \((x+m)(x+n)\) and \((x-m)(x-n)\) where \(m\) and \(n\) are positive. This is so that arithmetic doesn’t get in the way of noticing the relationships between the numbers in factored form and the numbers in standard form. In the next lesson, students will encounter expressions of the form \((x+m)(x-n)\) and \((x-m)(x+n)\).

Note that this course includes only four lessons on transforming quadratic expressions from standard form to factored form. This is intentional. The goal is to help students see factored form conceptually, understand what it can tell us about the function, and use that knowledge in modeling problems, including problems where the zeros are not rational and algebraic factoring wouldn't help. Too much attention to the algebraic skill of factoring can obscure these underlying concepts.

Sometimes—including in later courses and beyond—expressions do need to be written in factored form. With the universal availability of computer algebra systems, however, there is less need to spend lots of time learning how to factor by hand.