The expression:
(x^2+h^2)(x^2-h^2)
can be written as
(1+m-p)x^4-mp
where h, m, and p are constants. What is one possible value of m.
So the answer is h^2, but how? The explanation Khan Academy gave me was confusing and not thoroughly explained. Need a second opinion. Thanks!
If you multiply out the original expression you get::
x^4+x^2h^2 - x^2h^2 - h^4 OR x^4 -h^4
The 2nd expression can be written: x^4 + mx^4 - px^4 - mp or x^4 + [(m-p)x^4 - mp]
Thus [(m-p)x^4 - mp] = -h^4 and (m-p) must =0 so m = plus or minus h^2
@Eagledad33 See, the (m-p)=0 is where I get lost. Why is that? Everything before and after that I understand, but at that point is where my problem is.
Thanks for the response.
If (m-p) is non-zero, you’d be adding extra x^4 to the party. Breaking out the 2nd expression and then setting it equal to the 1st expression as follows:
X^4 + [(m-p)x^4 - mp] = X^4 - h^4
Notice how having X^4 adding on each side of the equation allows us to eliminate them thus giving the result for -h^4 as follows:
[(m-p)x^4 - mp] = -h^4
Since there’s now no x^4 on the right side of the equation, the x^4 portion on the left side has to be equal to zero so we can say that (m-p) = 0.
Ohhhh, okay. Got it now. Thanks!
@Eagledad33
No problem!
@IsaacFuture A slightly alternate way is to realize the expression is just a difference of two squares:
(x^2 + h^2)(x^2 - h^2) = x^4 - h^4
In general, any expression of the form (A+B)(A-B) can be written as A^2 - B^2 and vice versa.
Once you get x^4 - h^4, proceed as @Eagledad33 did.