Lesson 12

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.

In this lesson, students begin by dividing a 3rd degree polynomial by \((x-1)\) 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 staying organized and providing a structure to think through the division, they are not required for polynomial division, and some students may reason about the division in different ways.

For example, when dividing \(2x^2+7x+6\) by \((x+2)\), students can think backwards to figure out what \((x+2)\) would have to be multiplied by in order to get \(2x^2+7x+6\). They can reason that the \(x\) term from \((x+2)\) would need to be multiplied by \(2x\) to yield \(2x^2\). Then, since the whole polynomial \((x+2)\) is being multiplied by \(2x\), this would also result in a \(4x\). The polynomial we’re trying to get has the term \(7x\), so \(3x\) must be added to the \(4x\) from the previous step. This means that \((x+2)\) must be multiplied by 3. Again, the 2 in \((x+2)\) is also multiplied by this factor, so we also get a constant term of 6. This matches the constant term of \(2x^2+7x+6\), so we know that \((x+2)\) divides it evenly. Looking back at the terms we multiplied by at each step, we can conclude that \(2x^2+7x+6=(x+2)(2x+3)\). 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 \((x-1)\) is a linear factor of the polynomial. Building on their previous work, students make a sketch of the original 3rd degree polynomial after they rewrite it as three linear factors.

Dividing polynomials can be confusing at first to students since the number of terms involved mean there are a lot of pieces students need to keep an eye on. An important part of this lesson is students sharing their reasoning for how they deduced each term of the division (MP3).