Rewriting Quadratic Expressions in Factored Form (Part 1)
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.
- Apply the distributive property to multiply two sums or two differences, using a rectangular diagram to illustrate the distribution as needed.
- Generalize the relationship between equivalent quadratic expressions in standard form and factored form, and use the generalization to transform expressions from one form to the other.
- Use a diagram to represent quadratic expressions in different forms and explain (orally and in writing) how the numbers in the factors relate to the numbers in the product.
- Let’s write expressions in factored form.
- I can explain how the numbers in a quadratic expression in factored form relate to the numbers in an equivalent expression in standard form.
- When given quadratic expressions in factored form, I can rewrite them in standard form.
- When given quadratic expressions in the form of $x^2+bx+c$, I can rewrite them in factored form.
In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.
The coefficient of \(x\) in the expression \(3x + 2\) is \(3\). The coefficient of \(p\) in the expression \(5 + p\) is 1.
In an expression like \(5x + 2\) the number 2 is called the constant term because it doesn't change when \(x\) changes.
In the expression \(5x-8\) the constant term is -8, because we think of the expression as \(5x + (\text-8)\). In the expression \(12x-4\) the constant term is -4.
The linear term in a quadratic expression (in standard form) \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, is the term \(bx\). (If the expression is not in standard form, it may need to be rewritten in standard form first.)
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