This is the first of two lessons whose purpose is to introduce students to polynomial division, focusing specifically on dividing by linear factors. Up until this point, students have added, subtracted, and multiplied polynomials, but any types of division have been restricted to rewriting quadratics as the product of two linear factors.

Students begin by considering a series of equations and diagrams that show how 2 expressions written in factored form and standard form are equivalent. Next, they divide a 3rd-degree polynomial by using their understanding of the distributive property. It is important to note that while the diagrams used in this lesson and the next are useful for organizing work and providing a repeatable structure to think through the division (MP8), they are not required for polynomial division, and some students may reason about the division in different ways.

For example, when dividing by , students can think backward to figure out what would have to be multiplied by in order to get . They can reason that would need to be multiplied by to yield . Then multiplying the whole polynomial  by  results in . The polynomial we’re trying to get has the term , so must be added to the from the previous step. This means that must be multiplied by 3. Again, the whole polynomial is multiplied by this factor, resulting in plus a constant term of 6. This matches the constant term of , so we know that divides it evenly. Looking back at the terms we multiplied by at each step, we can conclude that . A diagram like the ones shown in this lesson is a compact way of keeping track of this reasoning.

Regardless of the division strategy used, the important takeaway for students is that when the division works out with no extra terms, we prove by example that the given polynomial is in fact a factor of the original polynomial. Building on previous work, students make a sketch of the original 3rd-degree polynomial after they rewrite it as three linear factors.