$(a+b)(a-b) = (a*a)+(a*-b)+(b*a)+(b*-b)$$= a^2-ab+ab-b^2 = a^2-b^2$Since we know that $(a + b)(a − b) = a^2 − b^2$, we need only turn the equation around to find the factorization form:$a^2 − b^2 = (a + b)(a − b)$Note: a perfect square is a term that is the square of another term. 4 is a perfect square of 2, 9 is a perfect square of 3, a2 is a perfect square of aThe factorization form says that we can factor $a^2 − b^2$, the difference of two squares, by finding the terms that produce the perfect squares and substituting these quantities into the factorization form.
Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.
The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible.The main motivation to decompose a rational function into a sum of simpler fractions is to make it simpler to perform linear operations on the sum.
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