I've just thought of this.

During a game of Scrabble someone puts down, 'anertious'.

The other players shake their head, look it up in the Scrabble dictionary. It's not there. Nor is it in any othere dictionary.

They all agree that it's not a word.

But the player is a writer. And he weaves the anecdote into a short story, which gets published.

Is it now a word?

]]>Btw, I can get exactly the same image with my 'old' (2021) Geogebra Classic 5 (Classic 6 is 2023).

My earliest use of Geogebra was in 2012, but I don't know which version or how capable it was.

]]>Thank you very much ]]>

Welcome to the forum.

You can simplify powers by using the rules for indices. https://www.mathsisfun.com/algebra/exponent-laws.html

phrontister has tidied up your maths notation to make the expression clearer. But is that what you meant?

Another interpretation of what you've posted is:

Please post again, making it clear which is right.

When you've got x and times in an expression you can use a . for the times to avoid confusion.

Bob

]]>1. All governments are corrupt.

I am afraid, this is a natural fact (by design). Therefore, nothing and no one can change it.

2. In the United States, the government has been on a mission to dump down its citizens for many decades. Why not? The dumber society is the easier to control.

As far as I know, this tactic has been also followed in all other countries since many decades ago. I noticed it too in my country.

4. As I age, solving math problems is my suduko puzzle, it's my cross word puzzle; it keeps my brain cells alive. No pun intended.

Although my private business is almost paralyzed since after year 2011 due to world's events and regulations, I keep designing new products (that needs lots of math) just to keep my brain cells alive.

5. To be honest with you, my circle of friends is very small for three reasons:

You are fortunate for having a very small circle of friends. My circle has zero radius

There is no one around me lately who enjoys math and education. It seems almost all people are attracted by other things.

About trust, two humans cannot decide to trust each other really before they discover each other while facing real hard times together.

ALL GOVERNMENTS do not want it's people to be well-informed. Like I said before, the dumber we are, the easier to control and brainwash. ALL GOVERNMENTS are bad, corrupt and not trustworthy.

Any government has to be run by rich persons or serving rich ones. Naturally, they have to protect each other, besides being protected by their masters (working behind the scenes). This was simplified 2000 years ago when it was said that a rich man is not supposed to enter ...

]]>(1/f) = (n - 1)[(1/a) + (1/b)]

]]>[2(x^2 + 9) - 2x(2x - 5)]/(x^2 + 9)^2

]]>3x^2 - 27 and 2x^2 - x - 15

I know that 3x^2 - 27 factors put to be

3(x^2- 9) = 3(x - 3)(x + 3).

I also know that 2x^2 - x - 15 factors out to be

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

I now have to select from the following:

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

LCM = 3(x - 3)^2 (x + 3) (2x + 5)

Do you agree?

]]>A. x^2 - x - 12 and x^2 - 8x + 16

x^2 - x - 12 factors out to be (x - 4)(x + 3).

x^2 - 8x + 16 factors put to be (x - 4)(x - 4) = (x - 4)^2.

LCM = (x + 3)(x - 4)^2

Is this correct?

]]>Thanks.

What edit did you make?

I changed the title and removed one of the questions.

]]>A. x^6 + 2x^3 + 1

B. 3 - 27x^2

Question A

Note: x^6 = (x^3)^2

Let u = x^3

I now get u^2 + 2u + 1.

(u + 1)(u + 1) = (u + 1)^2

Back-substitute for u.

(x^3 + 1)^2 = (x^3 + 1)(x^3 + 1).

The expression x^3 + 1 can be factored using the sum of cubes.

Doing so, I get (x + 1)(x + 1)(x^2 - x + 1)(x^2 - x + 1).

Answer: (x + 1)^2 (x^2 - x + 1)^2

Do you agree?

Question B

This expression can be factored using the difference of cubes.

3(1 - 9x^3)

The difference of cubes:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Let a = 1

Let b = 3x

3(1 - 3x)(1 + 3x + (3x)^2)

3(1 - 3x)(1 + 3x + 9x^2)

Do you agree?

]]>A. x^2 + 7

B. x^4 - 1

Question A

The polynomial x^2 + 7 cannot be factored. It is prime.

Question B

x^4 - 1

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

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

Note: x^2 + 1 is prime and thus cannot be factored.

Do you agree?

]]>