expand 1 power 4 minus 625 p power 4
Answers
Step-by-step explanation:
Summary: The commonest algebra mistake is probably rewriting A²+B² as (A+B)². “You can’t factor the sum of two squares on the reals!” your teacher tells you. While that’s generally true, there are some interesting exceptions.
Contents:
Sophie Germain’s Identity
Higher Powers
Complex Numbers
What’s New
Since Euler’s time at least, generations of students have tried to “factor” A²+B² as (A+B)². The lure of this siren song is so strong that I see even calculus students commit this blunder. Generations of teachers have sighed despairingly and tried to get students to remember that a sum of two squares can’t be factored on the reals.
When explaining Solving Polynomial Equations, I myself made that bare statement. But in correspondence in September 2009, Steve Schwartzman convinced me that I should say more. He proposed a sentence or two, but on the principle that a thing worth doing is worth overdoing, ...
It’s true that you can’t factor A²+B² on the reals — meaning, with real-number coefficients — if A and B are just simple variables. But if A and B have internal structure, the expression may be factorable after all, if you can find some other pattern. So it’s still true that a sum of squares can’t be factored as a sum of squares on the reals.
This page looks at some of the cases where a sum of squares can be factored using other techniques.
Sophie Germain’s Identity
The counterexample that Steve Schwartzman sent me in September 2009 is, as he told me, a form of Sophie Germain’s identity:
x4 + 4y4 = (x² + 2y² + 2xy) (x² + 2y² − 2xy)
Can you generalize this to a class of factorable sums of squares? Yes, you can.
Notice that the factors have the form of (P+Q)(P−Q), which of course multiplies to P²−Q². This suggests that, for factoring A²+B², it might be fruitful to look at (A+B)² minus something. That’s all well and good, but minus what?
The key is that (A+B)² = A²+2AB+B². Comparing that to A²+B², you see that there’s an extra term of 2AB. So you have
A² + B² = (A+B)² − 2AB
That right-hand side is factorable as a difference of squares, if 2AB is a perfect square. And that’s our factorization:
A² + B² = (A + B + √(2AB)) (A + B − √(2AB))
This identity is always true, but it’s useful for factoring only when 2AB is a perfect square.
More specifically, 2AB must be a perfect square if you want your factors to have rational coefficients. If you allow non-rational factors, you can factor more sums of squares, and if you allow complex factors you can factor any sum of squares.
Example 1: Factor 4x4 + 625y4.
Solution: Let A = 2x² and B = 25y²; then 2AB = 100x²y² is a perfect square and √(2AB) = 10xy.
4x4 + 625y4 = (2x² + 25y² + 10xy) (2x² + 25y² − 10xy)