$a^4-2 a^3 b+a^2 b^2-b^4$
Answer (1)
Recognize the first three terms as part of a perfect square.
Rewrite the expression as a difference of squares.
Factor the difference of squares.
The factored expression is ( a 2 − ab − b 2 ) ( a 2 − ab + b 2 ) .
Explanation
Understanding the Problem We are given the expression a 4 − 2 a 3 b + a 2 b 2 − b 4 and we want to factor it.
Completing the Square Notice that the first three terms look like a perfect square. Let's rewrite the first three terms: a 4 − 2 a 3 b + a 2 b 2 = ( a 2 ) 2 − 2 ( a 2 ) ( ab ) + ( ab ) 2 = ( a 2 − ab ) 2 .
Rewriting the Expression Now we can rewrite the entire expression as ( a 2 − ab ) 2 − b 4 .
Recognizing Difference of Squares We can recognize this as a difference of squares, since b 4 = ( b 2 ) 2 . So we have ( a 2 − ab ) 2 − ( b 2 ) 2 .
Factoring the Expression We can factor the difference of squares as ( a 2 − ab − b 2 ) ( a 2 − ab + b 2 ) .
Final Answer Therefore, the factored expression is ( a 2 − ab − b 2 ) ( a 2 − ab + b 2 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in calculus and other advanced mathematics. For example, in physics, you might encounter a situation where you need to find the roots of a polynomial to determine the equilibrium points of a system. Suppose the potential energy V ( x ) of a particle is given by V ( x ) = x 4 − 2 x 3 + x 2 − 1 . To find the equilibrium points, you would need to find where the force F ( x ) = − V ′ ( x ) is zero. This involves differentiating V ( x ) and then factoring the resulting polynomial to find its roots, which represent the equilibrium points. Factoring helps simplify complex expressions and solve equations more easily.
