MHB Is FOIL used to multiply radicals in sqrt{x - 3} • sqrt{x}?

[mathdad]

The picture shows a simple problem. However, my question has to do with multiplication of radicals.

I know how to use FOIL.

sqrt{x - 3} • sqrt{x} is slightly confusing.

Do I multiply radicand times radicand?

My question is:

Does sqrt{x - 3} • sqrt{x} become sqrt{x^2 - 3x} in the FOIL process?

View attachment 7895

 

[Greg]

You're not being clear about how you are applying FOIL but $\sqrt x\sqrt{x-3}=\sqrt{x^2-3x}.$

 

[tkhunny]

The picture shows a simple problem. However, my question has to do with multiplication of radicals.

I know how to use FOIL.

sqrt{x - 3} • sqrt{x} is slightly confusing.

Do I multiply radicand times radicand?

My question is:

Does sqrt{x - 3} • sqrt{x} become sqrt{x^2 - 3x} in the FOIL process?

#1 Please forget you EVER heard of "FOIL". That silliness works ONLY with two binomials and simply has no other purpose. Some poor students ever refer to "reverse-foiling". This is madness. Just learn to multiply. That is WHY you found FOIL slightly confusing. You weren't using it on two binomials. This is exactly how it confuses FAR MORE students than it helps.

#2 $\sqrt{x-3}\cdot\sqrt{x}$ may not be quite as easy as that. Here's why...

$\sqrt{x^2} = |x|$

$\sqrt{x}\sqrt{x} = x$

You tell me why and you will have answered your own question.

 

[mathdad]

#1 Please forget you EVER heard of "FOIL". That silliness works ONLY with two binomials and simply has no other purpose. Some poor students ever refer to "reverse-foiling". This is madness. Just learn to multiply. That is WHY you found FOIL slightly confusing. You weren't using it on two binomials. This is exactly how it confuses FAR MORE students than it helps.

#2 $\sqrt{x-3}\cdot\sqrt{x}$ may not be quite as easy as that. Here's why...

$\sqrt{x^2} = |x|$

$\sqrt{x}\sqrt{x} = x$

You tell me why and you will have answered your own question.

Wish I could tell you what the difference is but I am so into the FOIL method. It's what I was taught back in high school in the 1980s.

 

[tkhunny]

Wish I could tell you what the difference is but I am so into the FOIL method. It's what I was taught back in high school in the 1980s.

Hey, if you have two binomials, you can FOIL all you want. What's your plan if you've one binomial and one trinomial? How about three binomials? Eventually, you'll have to abandon it, no matter how hard you try to hold on. Don't make excuses for bad teaching methods that we've already seen cause you confusion. When I first encountered FOIL, I think it was in the 8th Grade, I recall thinking "That's stupid. Why wouldn't I just multiply them? I've been multiplying polynomials since the 2nd grade." (Like 452.3 * 25.75 = ??) It's probably one of the things that got me into trouble that year - teaching other students how to multiply, rather than the rote memorization with extremely limited usage. There is a Distributive Property of Multiplication over Addition for ALL such operations.

What do we know about the nature of 'x' if $\sqrt{x}$ exists?

What do we know about the nature of 'x' if $\sqrt{x^{2}}$ exists?

 

[mathdad]

Hey, if you have two binomials, you can FOIL all you want. What's your plan if you've one binomial and one trinomial? How about three binomials? Eventually, you'll have to abandon it, no matter how hard you try to hold on. Don't make excuses for bad teaching methods that we've already seen cause you confusion. When I first encountered FOIL, I think it was in the 8th Grade, I recall thinking "That's stupid. Why wouldn't I just multiply them? I've been multiplying polynomials since the 2nd grade." (Like 452.3 * 25.75 = ??) It's probably one of the things that got me into trouble that year - teaching other students how to multiply, rather than the rote memorization with extremely limited usage. There is a Distributive Property of Multiplication over Addition for ALL such operations.

What do we know about the nature of 'x' if $\sqrt{x}$ exists?

What do we know about the nature of 'x' if $\sqrt{x^{2}}$ exists?

For sqrt{x}, x must greater than or equal to 0.

For sqrt{x^2}, x can be any integer.

 

[Greg]

For sqrt{x^2}, x can be any integer.

Actually, $x$ can be any real number.

 

[tkhunny]

Okay, so what does that tell about whether \(\sqrt{x-3}\cdot\sqrt{x} = \sqrt{x^2 - 3x}\)?

 

[mathdad]

Okay, so what does that tell about whether \(\sqrt{x-3}\cdot\sqrt{x} = \sqrt{x^2 - 3x}\)?

It tells me that when two radicals with different radicands are multiplied, the product can be placed in one radicand.

So, sqrt{a} • sqrt{b} = sqrt{a•b}, as a general example.

 

[Prove It]

It tells me that when two radicals with different radicands are multiplied, the product can be placed in one radicand.

So, sqrt{a} • sqrt{b} = sqrt{a•b}, as a general example.

With the same domain as the original implied domains!

 

[mathdad]

Great.