Lesson 93 – Doodles Do Algebra

lesson93doodlecatcToday your child learns to find the greatest common divisor between two polynomials. The method is called the Euclidian Algorithm, but that part is not very important for your child. He just needs to learn how to do it.

There are 3 rules:

1. Divide the larger polynomial by the smaller one.

2. Ignore any factors of one polynomial if they are not common to the other polynomial. Unique factors are meaningless to finding the greatest common factor

3. Reserve the quantities of the greatest common factor as you find them during the division process.

The overall general idea is much the same – it is a matter of reducing each polynomial to its prime factors and then comparing, just a bit more complex since we are dealing with polynomials.

Download Lesson 93 Here


1. First figure out which polynomial is the larger one. 5a^2+5ax is definitely larger than a^2-x^2 because the first polynomial adds to 5 times the square of a and the second polynomial subtracts from the square of a (this is assuming a will be a positive number).

Second ignore factors of the smaller polynomial that are not common to the larger one. The first polynomial has factors 5, a, and (a+x). The second polynomial has factors (a+x) and (a-x). So we ignore the 5 and a.

Third step: the greatest common divisor between the two polynomials is then (a+x) because that is the common prime.

2. First figure out which polynomial is largest: in this case it is x^2+2cx+c^2 because the other polynomial subtracts a quantity.

Second, ignore the factors of one polynomial not common to the other. The first polynomial, x^2+2cx+c^2 , has factors which are (x+c), and (x+c). The second polynomial, x^3-c^2x , has factors x, (x+c), and (x-c).

So finally, the greatest common divisor is (x+c).

3. First, the largest of the two polynomials is x^2+5x-6.

Second, the first polynomial, x^2+5x-6, has factors which are (x+6), and (x-1). The second polynomial, x^2+2x-3, has factors which are (x+3), and (x-1).

So finally, the greatest common divisor is (x-1).

See, it is not too hard if you just keep the overall idea in mind and work the steps.


Lesson 92 – Doodles Do Algebra

lesson92doodlepoodlecToday your child begins learning about the greatest common divisor. This is the largest shared combination of prime factors that make up two expressions.

As an example, the greatest common divisor between 6ab and 15a^2c is 3a. For more complicated expressions, you simply find all the prime factors of each expression and then select all those in common between both expressions.



1. Step One is find the prime factors. So prime factors of 4a^2x^3 are 2, 2, a, a, x, x, x. And the prime factors of 10ax^3 are 2, 5, a, x, x, x. Comparing the two, the common primes are 2, a, x, x, x so the greatest common divisor is 2*a*x*x*x=2ax^3

2. Prime factors of 9abc^3 are 3, 3, a, b, c, c, c and prime factors of 12bc^4x are 2, 2, 3, b, c, c, c, c. So comparing the two, the common primes are 3, b, c, c, c and the greatest common divisor is 3bc^3

3. Prime factors of 3a^4y^3 are 3, a, a, a, a, y, y, y. Prime factors of 6a^5x^3y^5 are 2, 3, a, a, a, a, a, x, x, x, y, y, y, y, y. Prime factors of 9a^5y^4z are 3, 3, a, a, a, a, a, y, y, y, y, z. So comparing the three expressions, the common primes are 3, a, a, a, a, y, y, y and the greatest common divisor is 3a^4y^3.

4. Prime factors of 4a^3b^2x^5y^3 are 2, 2, a, a, a, b, b, x, x, x, x, x, y, y, y. Prime factors of 8a^5x^3y^3 are 2, 2, 2, a, a, a, a, a, x, x, x, y, y, y. So comparing the two expressions, the common primes are 2, 2, a, a, a, x, x, x, y, y, y and the greatest common divisor is 4a^3x^3y^3.

5. 4x^3

6. 3a^3y^3


Lesson 91 – Doodles Do Algebra

lesson91doodleboyToday is the last day of factoring. DoodleTwo explains that factoring is simply a way of shortening the work and simplifying your problems because you can cancel common factors, especially when you are working with fractions.

First write the problem out and then factor what you can, just like in the example worked out on the worksheet. Then you cancel common factors.

And that is exactly what your child is doing today.



1. (x^2+1)/(x+1)*(x^2-1)=(x^2+1)(x+1)(x-1)/(x+1)=(x^2+1)(x-1)

2. (x^2-5x+6)*(x^2-7x+12)/(x^2-6x+9)=(x-2)(x-3)(x-3)(x-4)/((x-3)(x-3))=(x-2)(x-4)


Lesson 90 – Doodles Do Algebra

lesson90doodlegirl These in the next few lessons, we finish off the ideas of factoring and begin to look at the greatest common factor.

Today your child learns how to separate a quadratic trinomial into its factors. DoodleOne explains it very nicely, so I will let her do it on the worksheet.

If your child has trouble, just work through a few with her and show her that the process is kind of like working out ciphers using a couple of clues. You are given two numbers, one that represents a times b, and the other that represents either a plus b or a minus b or b minus a. Then you have to work out values for a and b are and that tells you what the factors of the quadratic trinomial. It all sounds much more complicated than it really is.



1. (x+2)(x+3) The idea here is to think of possible factors for 6 (like 2*3 or 6*1) and then decide if either of those possiblilities can be combined to make 5. Since 2+3 is 5, then you get your factors for the equation.

2. (a+3)(a+4) So here you think of ways to factor 12 (like 12*1, 2*6, or 3*4) and then figure out which set combines to make 7. 3+4 is 7, so the values for a and b are 3 and 4.

3. (x-2)(x-3) Here you have to find factors for 6, keeping in mind that they mst combine to a negative 5. Following the same process as the last 2 problems, you decide that -2*-3 are the best factors because they add to -5 and multiply to yield a +6.

4. (x-10)(x+1)

5. (x+3)(x-2)

6. (x+2)(x-1)

7. (x-8)(x-5)

8. (x-8)(x+1)

9. (x-9)(x+2)

10. (x-6)(x+5)

11. (3)(x^2+4x-5)=3(x-5)(x+1)


Lesson 89 – Doodles Do Algebra

lesson89doodlepigcToday is more practice in factoring polynomials with two more rules to learn: divisibility rules for a^4-b^4 and a^3+b^3, and also how to abstract those rules to higher powers of a and b.



1. (m+n)(m+n)

2. (a-bx)(a-bx)

3. (2x-5z)(2x-5z)

4. (x+y)(x-y)

5. (3m+4n)(3m-4n)

6. (ab+cd)(ab-cd)

7. (x)(a^2-x^2)=(x)(a+x)(a-x)

8. (y+1)(y^2-y+1)

9. (x-1)(x^2+x-1)

10. (2a-3b)(2^2a^2-4a*3b-3^2b^2)=(2a-3b)(4a^2-12ab-9b^2) here you need to recognize that 8 is 2 cubed and 27 is 3 cubed and then it all can be treated as x cubed minus y cubed and your can apply the general formula from lesson 88

11. (a+b)(a^4-a^3b+ab^3-b^4)

12. (a+b)(a-b)(a^4-a^2b^2+b^4)

these last few problems require a bit of thinking power and some long division, but other than that they are not hard.


Lesson 88 – Doodles Do Algebra

lesson88doodlecatcToday your child expands the idea of factoring to a product of 2 or more polynomials.

This is the old a^2+2ab+b^2=(a+b)(a+b) idea that you remember for high school algebra.

The important point here is that you are factoring a composite into two primes.

DoodleCat explains it quite well on the worksheet and expands the idea to a general equation that relies on your child knowing how to divide polynomials. But the advanced part is just a preview for today and the problems are fairly simple separation of a polynomial into its simplest factors.

When I went through this part with my own children, I wrote the formulas out on very large papers and taped them to the pocket doors that separate our living room from our dining room. That way my kids could refer to them and have them around to stare at while waiting to be excused from the table after dinner.



1. (x+y)(x+y)

2. (3a+2b)(3a+2b)

3. (2+3x)(2+3x)


Lesson 87 – Doodles Do Algebra

lesson87doodlepoodlecToday your child learns how to separate algebraic expressions into factors that include the greatest monomial that evenly divides the expression and its quotient.

It may sound complex, but it really isn’t – especially since your child has now worked through 86 lessons that have included learning about the vocabulary of algebra.

If she has a little trouble getting started just walk her through the first couple of problems and she will quickly see the pattern.



1. a*(m+c)

2. bc*(c+d)

3. 2x*(2x+3y)

4. 3xy*(2ax+3by-4cx)

5. 5ax^2*(1-7y+ay)

6. 7a^2xy*(2ax+3xy^2-5ay)

7. 3bc^2*(2x-5c-b^2c)

8. a^2cm^2*(am+c-am)=a^2cm^2*(c)


Doodles Do Algebra – Lesson 86

lesson86doodleboyToday your child expands his knowledge of factoring numbers to algebraic terms. DoodleTwo explains it very well on the worksheet, so I think no additional explanation is necessary.



1. 15a^2bc=3*5*a*a*b*c

2. 21ab^2d=3*7*a*b*b*d

3. 35abc^2x=5*7*a*b*c*c*x

4. 39a^2m^2x=3*13*a*a*m*m*x


Lesson 85 – Doodles Do Algebra

lesson85doodlegirlToday we practice with prime numbers and composite numbers in preparation for factoring.

It is not too hard, especially once your child completes the table of prime numbers less than 100 at the top of the page.



Primes under a 100: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

Prime factors of 105: 5, 3, 7

Prime factors of 210: 2, 5, 3, 7

Resolve 4290 into its prime factors: 2, 3, 5, 11, 13


Lesson 84 – Doodles Do Algebra

lesson84doodlepigcThis is the last of the copywork days for now. Next week we begin factoring based on an understanding of these theorems.