Update: This Lesson Is Part of Book 3 of Doodles Do Algebra: “The Basic Math of Algebra” available on Amazon.
Available in the book, “The Basic Math of Algebra, Book 3 of Doodles Do Algebra“
This week your child begins learning how to divide one polynomial by another. Taking the ten thousand foot view at progress so far, we have taught him how to understand and mentally solve algebraic problems, how to manipulate algebraic phrases (monomials and polynomials) using addition, subtraction, multiplication, and now we finish up with division. After that your child is ready to learn how to solve simple equations (one equation and one unknown as well as two equations and two unknowns and three equations and three unknowns), followed by some time learning the basic math of manipulating powers and exponents, and then on to solving higher order equations. By now I think you can probably see the overall method behind this way of learning algebra: everything is derivable and your child learns all the tools and techniques to make those derivations. There is very little, “memorize this formula and learn to apply it under specific circumstances,” because I (and the great old mathematicians that wrote the standard textbooks of the 1700s and 1800s believe that learning algebra is being able to do algebra, not apply a formula out of a book.
So today your child learns to divide one polynomial by another. Basically this is a process my kids called the long division algebra waltz.
In a broad picture, you figure out the contribution of terms of the largest power for each of the variables you have in your divisor (in this case the a term and the x term). The left hand area of the paper is for keeping track of how much of the division you have left to take care of (answers to multiplying and subtracting) and the right hand side of the paper is for keeping the terms of the final answer as your write them down. It is not a hard dance, once you see the pattern and get used to it.
And here are the steps:
0. rewrite the problem with an upside down division box, as shown on the worksheet
1. look at the first term of the divisor. Ask yourself how many times do you have to multiply it to make it match the term in the divided that has the highest power of the variable that is in first term of the divisor and write it in the box below the first explanation (#1) (in our example today the first term in the divisor is 2a, so the variable in that term is a and you have to multiply 2a by 3a to get the term in the divisor, which is the term)
2. now you multiply that term in the box by the whole divisor and write that in the space below the #2 shoe (we are laying out the steps to the long division algebra waltz) and the answer is
3. This is the contribution which 3a makes to the total dividend, so just like long division in mathematics, we subtract that contribution from the original dividend. So . This may look messy at this point, but not to worry as we continue on things will look simpler, and hey – we just took care of the term.
4. Now our long division algebra waltz steps us back over to look at the original divisor and we ask what do you have to multiply the second term by to make it equal to the term in the dividend with the highest power of x in the answer from step 3? Basically you have accounted for the contribution of the first term (2a) and subtracted it out so now you deal with the answer from step 3 as the dividend. And so the number you write in the cloud bubble on the worksheet is -2x since -2x times -3x equals .
5. Only two steps left. First is to find the contribution your new term in the divisor makes to the answer, just like you did in step 2. So you multiply .
6. Now you subtract that from the answer you got in step 3 when you subtracted the contribution of 3a to the answer. So -4ax+6x^2-(-4ax+6x^2)=0. Since you got a 0, you are done. You found all the contributions and divided by and got . If you want to double check the answer, just multiply and you will find that it is correct.
Awesome job, mom! And if your kids are not too terribly confused this first time through, even if you walked them through it step by step and wrote everything out yourself, call it a win! They are going to get a lot more practice and the goal this first time through is just to explain and show them. Also don’t worry if you don’t ever remember doing this in algebra class in high school. I don’t either, although my husband did do it, so somewhere in the 1960’s or 1970’s teachers in California stopped teaching it and resorted to the list of reference formulas for factoring a limited class of polynomials. This way is soooo much better and easier to understand overall. It takes the mystery straight out of algebra.