Math Is Fun Forum

amnkb wrote:

All the factors are:
1, 2, 3, 4, 6, 12, 1x, 2x, 3x, 4x, 6x, 12x, 1x^2, 2x^2, 3x^2, 4x^2, 6x^2, 12x^2

If I understood well the exercise, amnkb gave the right answer.

For example. let us assume x=5
12x^2 = 300
So, its factors are:
1, 2, 3, 4, 6, 12, 5, 10, 15, 20, 30, 60, 25, 50, 75, 100, 150, 300
-----------------   -----------------------   -----------------------------

FelizNYC wrote:

Find the GCF of 45 and 75.

Another way is to do the prime factorization w/ a table:

factors
 45: 3 * 3 * 5
 75: 3     * 5 * 5
---+--------------
GCF: 3     * 5    = 15

x^2 - 2x + 1 + y^2 + 4y + 4 - 9 = 0

x^2 + y^2 - 2x + 4y + 1 + 5 - 9 = 0

The plus 4 on the correct line has become plus 5 on the next line.

Bob

FelizNYC wrote:

(x - 1)^2 + (y + 2)^2 = 9
...
x^2 + y^2 - 2x + 4y - 3 = 0

(x - 1)^2 + (y + 2)^2 = 9

Let us assume x=1:
(1 - 1)^2 + (y + 2)^2 = 9
(y + 2)^2 = 9
y + 2 = 3
y = 3 - 2
y = 1

Now,
x^2 + y^2 - 2x + 4y - 3 = 0

Let us verify if this final form is also satisfied for x=1 and y=1
1^2 + 1^2 - 2*1 + 4*1 - 3 = 0
1 + 1 - 2 + 4 - 3 = 1 ≠ 0

I am afraid that you did a typo somewhere.

FelizNYC wrote:

In that case, the equation should be

x^2 + y^2 + 10x + 4y - 20 = 0

Yes?

Yes.

Q1 Perfect.

Q2. Almost. Correct up to this line:

x^2 + 10x + 25 + y^2 + 4y + 4 = 49

You've brought the 49 across to the left hand side but forgotten there's already +25 and + 4 there.

Bob

Maybe the problem here is that the word factor is being used to mean two similar but different things.

If N is a whole number then another whole number m is a factor if N/m is a whole number.

eg. 8 is a factor of 48 because 48/8 = 6

But if an algebraic expression (E) can be split into two expressions (F and G) that multiply to make E then F and G are said to be factors.

eg. 

x and (x+2) are factors of E. 

If only numbers are involved then the GCF is clear,

eg The GCF of 45 and 75 is 15

In pure mathematical terms saying that the GCF of 12x^2 and 28x^4 is 4x^2 is a questionable use of GCF because we don't even know if x is a whole number so how can we ask questions about the greatest factor.

But the Prof. goes on to use his earlier method to factorise (second definition) algebraic expressions. Nothing wrong with that; methods that work but have use dubious logic do exist in maths . I teach integration by saying have a guess based on what you know about differentiation; check if it works; and modify your guess if it doesn't.

Bob

Watched it. Great vid!

But I didn't hear him say what he means by 'evenly divided'.  It's not a maths expression that I'm familiar with.

Did it come from a text book question?

Bob

Relatively prime means the two (or more) numbers have only 1 as a common factor.

For example: 6(= 2 x 3) and 11 are relatively prime, 45( = 3 x 3 x 5) and 77( = 7 x 11) are relatively prime

6 and 15 are not;  as 3 is also a factor of both.

If you factorise out the GCF from an expression then the numbers left must have no common factor because if there was another factor for both then you haven't found the greatest common factor.

So the numbers left are relatively prime.

I'll make up another example.

42x + 70 = 2(21x + 35)  21 and 35 are not relatively prime, but I haven't yet factored out all the common factors.

Let's do that:

42x + 70 = 2(21x + 35) = 14(3x + 5)  Now we have 3 and 5, and these are relatively prime.

Bob

Correct.

Bob