Today 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 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.

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1. Step One is find the prime factors. So prime factors of are 2, 2, a, a, x, x, x. And the prime factors of 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. Prime factors of are 3, 3, a, b, c, c, c and prime factors of 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

3. Prime factors of are 3, a, a, a, a, y, y, y. Prime factors of are 2, 3, a, a, a, a, a, x, x, x, y, y, y, y, y. Prime factors of 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 .

4. Prime factors of are 2, 2, a, a, a, b, b, x, x, x, x, x, y, y, y. Prime factors of 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 .

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Today 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.

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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.

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1. 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. 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. 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.

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Today is more practice in factoring polynomials with two more rules to learn: divisibility rules for and , and also how to abstract those rules to higher powers of a and b.

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10. 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

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

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

This is the old 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.

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Today 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.

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Today 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.

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Today 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.

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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

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

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