Dividing Polynomials – Script SCMP Summer 2008
Dividing Polynomials Script
Use an example problem, such as (2x2 + 6x + 4) divided by (x + 2) and ask students to build a rectangular array with tiles representing the first expression (dividend) so that the array has a height equal to the second expression (divisor). Allow students time to discover the task is possible without using neutral pairs or having any tiles left over. Discuss what this means in terms of the division. Give the students another example, like (x2 + 3x + 5) divided by (x + 2), and direct them to build another array with tiles representing the first expression (dividend) having a height equal to the second expression (divisor). Let them discover this is not possible unless they have leftover tiles. Show them that this will be the remainder and can be justified as such by multiplying the divisor by the quotient and noting that adding the remainder will give the correct dividend. This shows the connection between division and multiplication that they are inverse operations and one “undoes” the other.
Give students a third example, such as (x2 + 2x – 5) divided by (x – 2) and allow them time to explore the possibility of building an array with tiles representing the first expression (dividend) having a height equal to the second expression (divisor). Discuss the effect the negative terms have on the array. When students have had time to try the problem, discuss the difficulties they had and how it was necessary to add neutral pairs to complete the array without changing the value of the dividend. Give students a fourth example, such as (2x2 - 3x – 5) divided by (x + 1), and allow time for the students to investigate. They will discover that adding two neutral pairs will complete the array and the quotient will have no remainder. Discuss how students may check their solutions using multiplication of the quotient and divisor to produce the dividend.
Allow the time to investigate the six practice problems and ask different people to share their models and solutions with the group. Encourage the presenters to clearly explain the use of neutral pairs and negative area in completing the arrays.
Discuss how this model for division of polynomials connects to division of whole numbers and multiplication and division of integers. Discuss implications for instruction.
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