Factoring Question

[Jason76]

How does

\(\displaystyle \sqrt{1 - 9x^{2} + 24x^{4} - 16x^{6}}\)

become

\(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\) ?

How does

\(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\)

become

\(\displaystyle \sqrt{(4x^{2} - 1)^{2}(1 - x^{2})}\)?

 

[mmm4444bot]

It's called "factoring", Jason. A concept you have yet to learn.

 

[Deleted member 4993]

How does

\(\displaystyle \sqrt{1 - 9x^{2} + 24x^{4} - 16x^{6}}\)

become

\(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\) ?

How does

\(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\)

become

\(\displaystyle \sqrt{(4x^{2} - 1)^{2}(1 - x^{2})}\)?

\(\displaystyle \sqrt{1 - 9x^{2} + 24x^{4} - 16x^{6}}\)

substitute:

u = x²

Then we have:

\(\displaystyle \sqrt{1 - 9u + 24u^{2} - 16u^{3}}\)

Now use rational root theorem and remainder theorem to see that u = 1 is a root of the given equation.

Continue......

#2)

\(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\)

Hint:

\(\displaystyle (4x^{2} - 1)^{2}\) = ??

Use FOIL

[/QUOTE]

 

[soroban]

Hello, Jason76!

How does \(\displaystyle \sqrt{1 - 9x^{2} + 24x^{4} - 16x^{6}}\) become \(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\) ?
This one is tricky.

\(\displaystyle 1 - 9x^2 + 24x^4 - 16x^6\)

. . \(\displaystyle =\;1 -x^2 - 8x^2 + 24x^4 - 16x^4\)

. . \(\displaystyle =\; (1 - x^2) - 8x^2(1 - 3x^2 + 2x^4)\)

. . \(\displaystyle =\; (1-x^2)-8x^2(1-x^2)(1-2x^2)\)

. . \(\displaystyle =\; (1-x^2)\big[1 - 8x^2(1-2x^2)\big]\)

. . \(\displaystyle =\; (1-x^2)(1 - 8x^2 + 16x^4)\)

. . \(\displaystyle =\;(1-x^2)(16x^4-8x^2+1)\)

How does \(\displaystyle \sqrt{(1 - x^{2})(16x^{4} - 8x^{2} + 1)}\) become \(\displaystyle \sqrt{(4x^{2} - 1)^{2}(1 - x^{2})}\) ?
But you should know this one.

Can't you see that: .\(\displaystyle 16x^4-8x^2+1 \:=\: (4x^2-1)(4x^2-1) \;=\;(4x^2-1)^2\)


I think mmm4444bot is right.
You don't seem to have mastered Factoring yet.
So how come they're giving you messy problems with square roots?

 

[mmm4444bot]

Soroban, I'm not sure that Jason is taking a class. He bought a book. His threads indicate an attempt to learn calculus by skipping algebra.

 

[Jason76]

Soroban, I'm not sure that Jason is taking a class. He bought a book. His threads indicate an attempt to learn calculus by skipping algebra.

I'll try to go back and learn more algebra before taking on Calculus. But I understand most Algebra; I just need some work on factoring.

One more thing, in Soraban's post.

He said:

\(\displaystyle (1 - x^{2}) - 8x^{2} (1 - x^{2})(1 - 2x^{2})\)

factors to

\(\displaystyle (1 - x^{2})[1 - 8x^{2}(1 - 2x)]\)

Where does the 2nd 1 in the 2nd line come from. The one before the

\(\displaystyle -8x^{2}\) ?

 

[mmm4444bot]

\(\displaystyle (1 - x^{2}) - 8x^{2} (1 - x^{2})(1 - 2x^{2})\)

factors to

\(\displaystyle (1 - x^{2})[\)\(\displaystyle 1\)\(\displaystyle - 8x^{2}(1 - 2x)]\)

Where does the 2nd 1 in the 2nd line come from.

It's called factoring, Jason.

Soroban factored (1 - x^2) out of the first line.

The 1 must be present in the second line; otherwise, expanding the second line does not result in the first line.

For example, factor k + k^2:

k(1 + k)


I think you're going about math learning backwards. I think that your are wasting of lot of your time. Why not start at the beginning and go forward versus starting toward the end and going backward?