Polynomials are expressions containing variables and integers using only arithmetic operations and positive integer exponents between them. All polynomials have a factored form where the polynomial is written as a product of its factors. All polynomials can be multiplied from a factored form into an unfactored form by using the associative, commutative and distributive properties of arithmetic and combining like terms. Multiplying and factoring, within a polynomial expression, are inverse operation. That is, one operation "undoes" the other.
Multiply the polynomial expression by using the distributive property until each term of one polynomial is multiplied by each term of the other polynomial. For example, multiply the polynomials x + 5 and x - 7 by multiplying every term by every other term, as follows:
(x + 5)(x - 7) = (x)(x) - (x)(7) + (5)(x) - (5)(7) = x^2 - 7x + 5x - 35.
Combine like terms in order to simplify the expression. For example, to simply the expression x^2 - 7x + 5x - 35, add the x^2 terms to any other x^2 terms, doing the same for the x terms and constant terms. Simplifying, the above expression becomes x^2 - 2x - 35.
Factor the expression by first determining the greatest common factor of the polynomial. For example, there is no greatest common factor for the expression x^2 - 2x - 35 so factoring must be done by first setting up a product of two terms like this: ( )( ).
Find the first terms in the factors. For example, in the expression x^2 - 2x - 35 there is a x^2 term, so the factored term becomes (x )(x ), since this is required to give the x^2 term when multiplied out.
Find the last terms in the factors. For example, to get the final terms for the expression x^2 - 2x - 35, a number is needed whose product is -35 and sum is -2. Through trial and error with the factors of -35 it can be determined that the numbers -7 and 5 meet this condition. The factor becomes: (x - 7)(x + 5). Multiplying this factored form gives the original polynomial.
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