how does k(2k+1)(2k-1)+(2k+1)^2 become (2k+1)(k(2k-1)+3(2k+1))?
First, ask whether the two expressions are in fact equal!
k(2k+1)(2k-1)+(2k+1)^2 = (2k+1)(k(2k-1)+3(2k+1))
only for k = -1/2
This step was wrong:
4k^3 - k + (4k^2 + 4k + 1)
= 4k^3 + 4k^2 - k + 4k^2 + 4k + 1
you pulled an extra 4k^2 out of thin air!
So, let's see what we really have:
k(2k+1)(2k-1) + (2k+1)^2
k(2k+1)(2k-1) + (2k+1)(2k+1)
(2k+1)(x(2k-1) + 2k+1)
(2k+1)(2k^2 + k + 1)
To expand the expression k(2k+1)(2k-1) + (2k+1)^2, we can distribute the terms:
k(2k+1)(2k-1) + (2k+1)(2k+1)
First, let's expand the first term:
k(2k+1)(2k-1) can be expanded as:
= k * (4k^2 - 1)
Now, let's expand the second term:
(2k+1)(2k+1) can be expanded using the foil method:
= 2k * 2k + 2k * 1 + 1 * 2k + 1 * 1
= 4k^2 + 2k + 2k + 1
= 4k^2 + 4k + 1
Now, let's substitute these expanded terms back into the original expression:
k(2k+1)(2k-1) + (2k+1)^2
= k * (4k^2 - 1) + (4k^2 + 4k + 1)
Next, we can distribute the k into (4k^2 - 1) using the distributive property:
= 4k^3 - k + (4k^2 + 4k + 1)
Finally, we can simplify the expression by combining like terms:
= 4k^3 + 4k^2 - k + 4k^2 + 4k + 1
= 4k^3 + 8k^2 + 3k + 1
Therefore, the given expression k(2k+1)(2k-1) + (2k+1)^2 simplifies to (2k+1)(k(2k-1) + 3(2k+1)).
You are correct, my previous response contained errors. Let's continue from where we left off:
The expression k(2k+1)(2k-1) + (2k+1)^2 simplifies to:
k(2k+1)(2k-1) + (2k+1)(2k+1)
= (2k+1)(x(2k-1) + 2k+1)
Now, let's simplify the expression within the parentheses:
x(2k-1) + 2k + 1 can be simplified by distributing the x:
= 2kx - x + 2k + 1
Substituting this back into the original expression:
(2k+1)(x(2k-1) + 2k+1)
= (2k+1)(2kx - x + 2k + 1)
Finally, we can distribute (2k+1) into the expression (2kx - x + 2k + 1):
= 2k(2kx - x + 2k + 1) + (2k+1)(2kx - x + 2k + 1)
= 4k^2x - 2kx + 4k^2 + 2k + 2kx - x + 2k + 1
Now, we can combine like terms:
= 4k^2x - x + 4k^2 + 4k + 1
Therefore, the correct simplification of the given expression is:
(2k+1)(2k^2 + k + 1)
= 4k^2x - x + 4k^2 + 4k + 1