In an earlier lesson, students transformed quadratic expressions from standard form into factored form. There, the factored expressions are products of two sums, \((x+m)(x+n)\), or two differences, \((x-m)(x-n)\). Students continue that work in this lesson, extending it to include expressions that can be rewritten as products of a sum and a difference, \((x+m)(x-n)\).
Through repeated reasoning, students notice that when we apply the distributive property to multiply out a sum and a difference, the product has a negative constant term, but the linear term can be negative or positive (MP8). Students make use of structure as they take this insight to transform quadratic expressions into factored form (MP7). They see that if a quadratic expression in standard form (with coefficient 1 for \(x^2\)) has a negative constant term, one of its factors must have a negative constant term and the other must have a positive constant term.
- Apply the distributive property to multiply a sum and a difference, using a diagram to illustrate the distribution as needed.
- Given a quadratic expression of the form $x^2+bx+c$, where $c$ is negative, write an equivalent expression in factored form.
- When multiplying a sum and a difference, explain (orally and in writing) how the numbers and signs of the factors relate to the numbers in the product.
- Let’s write some more expressions in factored form.
- I can explain how the numbers and signs in a quadratic expression in factored form relate to the numbers and signs in an equivalent expression in standard form.
- When given a quadratic expression given in standard form with a negative constant term, I can write an equivalent expression 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.)
zero product property
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.