Factoring and simplifying an expression

Hi, the problem is to factor and simplify the expression

$\displaystyle (x^2-4)(x^2+3)^{1/2} - (x^2-4)^2(x^2+3)^{3/2}$

I can factor the left side but the right leaves me stuck.

I tried multiplying both terms in the right and got $\displaystyle (x^4-8x^2+16) \sqrt{x^6+9x^4+27x^2+27}$

but I feel like I am going the wrong way and don't think it should be this complicated.

The answer in the back of the book is $\displaystyle (x+2)(x-2)(x^2+3)^{1/2}(-x^4 +x^2 +13)$

I don't know where the final term came from.

Thank you.

Re: Factoring and simplifying an expression

Quote:

Originally Posted by

**Foxlion** Hi, the problem is to factor and simplify the expression

$\displaystyle (x^2-4)(x^2+3)^{1/2} - (x^2-4)^2(x^2+3)^{3/2}$

I can factor the left side but the right leaves me stuck.

I tried multiplying both terms in the right and got $\displaystyle (x^4-8x^2+16) \sqrt{x^6+9x^4+27x^2+27}$

but I feel like I am going the wrong way and don't think it should be this complicated.

The answer in the back of the book is $\displaystyle (x+2)(x-2)(x^2+3)^{1/2}(-x^4 +x^2 +13)$

I don't know where the final term came from. Thank you.

$\displaystyle (x^2 - 4)(x^2 + 3)^{\frac{1}{2}}[1 - (x^2 - 4)(x^2 + 3)] $

$\displaystyle (x + 2)(x - 2) \sqrt{x^2 + 3} \ [1 - x^4 - 3x^2 + 4x^2 + 12] $

Now just combine alike terms in the brackets.

:)

Re: Factoring and simplifying an expression

where does the 1 come from?

Re: Factoring and simplifying an expression

Quote:

Originally Posted by

**Foxlion** where does the 1 come from?

$\displaystyle [{\color{blue}1} - (x^2 - 4)(x^2 + 3)]=[{\color{blue}1} - (x^4-x^2-12)] $

Re: Factoring and simplifying an expression

Hello, Foxlion!

Quote:

$\displaystyle \text{Factor and simplify: }\:(x^2-4)(x^2+3)^{\frac{1}{2}} - (x^2-4)^2(x^2+3)^{\frac{3}{2}}$

$\displaystyle \text{Book answer: }\:(x+2)(x-2)(x^2+3)^{\frac{1}{2}}(-x^4 +x^2 +13)$

$\displaystyle \text{If you had: }\,ab^{\frac{1}{2}} - a^2b^{\frac{3}{2}},\:\text{ can you see that:}$

. . $\displaystyle \text{both terms have a factor of }a,$

. . $\displaystyle \text{both terms have a factor of }b^{\frac{1}{2}}\,?$

$\displaystyle \text{Factor out }ab^{\frac{1}{2}},\:\text{ and we have: }\:ab^{\frac{1}{2}}(1 - ab)$

If you can't follow that, you need more help than we can provide.

Re: Factoring and simplifying an expression

no I get that, but how does

$\displaystyle (x^2-4)^2(x^2+3)^{3/2}$

become $\displaystyle 1-(x^2-4)(x^2+3)$ ?

Re: Factoring and simplifying an expression

yeah I see that, thank you.

Re: Factoring and simplifying an expression

Hello again, Foxlion!

Quote:

$\displaystyle \text{Factor and simplify: }\:(x^2-4)(x^2+3)^{\frac{1}{2}} - (x^2-4)^2(x^2+3)^{\frac{3}{2}}$

Okay, you understand this: .$\displaystyle ab^{\frac{1}{2}} - a^2b^{\frac{3}{2}} \:=\:ab^{\frac{1}{2}}(1 - ab)$ .[1]

The given problem is a messier version of this equation.

. . They used $\displaystyle (x^2-4)$ instead of $\displaystyle a$

. . and used $\displaystyle (x^2+3)$ instead of $\displaystyle b.$

Substitute those expression into [1].

Got it?

Re: Factoring and simplifying an expression

Quote:

Originally Posted by

**soroban** hello again, foxlion!

okay, you understand this: .$\displaystyle ab^{\frac{1}{2}} - a^2b^{\frac{3}{2}} \:=\:ab^{\frac{1}{2}}(1 - ab)$ .[1]

the given problem is a messier version of this equation.

. . they used $\displaystyle (x^2-4)$ instead of $\displaystyle a$

. . and used $\displaystyle (x^2+3)$ instead of $\displaystyle b.$

substitute those expression into [1].

Got it?

got it