Factor:
50r^8 – 32r^2
The caret symbol represents a power.
2r^2(25r^6+16)=
2r2(25r6+16)
DrBob222 forgot a minus sign:
50r^8 – 32r^2 =
2r^2(25r^6 - 16)
Then, inside the brackets, you see a difference of two squares:
25r^6 - 16 = X^2 - Y^2
with X = 5 r^3 and Y = 4
You can then simplify this using
X^2 - Y^2 = (X + Y)(X - Y) --->
25r^6 - 16 = (5 r^3 + 4)(5 r^3 - 4)
You can factor each of these factors further by finding one of the roots.
5 r^3 + 4 = 0 -->
r = -(4/5)^(1/3)
This means that you can divide
5 r^3 + 4 by [r + (4/5)^(1/3)] to obtain a quadratic term. In case of the other factor you find the root
r = (4/5)^(1/3) and you can thus divide the term by [r- (4/5)^(1/3)] to find a quadratic factor.
Thanks to Count Iblis for catching my error. I got so carried away with trying to make the exponents look good that I simply read a minus sign but typed a + sign. Such is old age, bad eyes, and an elevator that doesn't go all the way to the top.
To factor the expression 50r^8 - 32r^2, we can look for common factors first. In this case, we see that both terms have a factor of 2r^2. We can factor this out:
50r^8 - 32r^2 = 2r^2(25r^6 - 16)
Now, we can focus on factoring the expression inside the brackets, 25r^6 - 16.
This expression can be rewritten as the difference of two squares:
25r^6 - 16 = (5r^3)^2 - 4^2
Using the formula for the difference of two squares, we have:
(a^2 - b^2) = (a + b)(a - b)
In this case, our a is 5r^3 and our b is 4. Substituting these values into the formula, we get:
25r^6 - 16 = (5r^3 + 4)(5r^3 - 4)
So, the fully factored form of the expression 50r^8 - 32r^2 is:
2r^2(5r^3 + 4)(5r^3 - 4)