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In the last two types of problems (Sections [link] and [link] ), we knew one of the factors and were able to determine the other factor through division. Suppose, now, we’re given the product without any factors. Our problem is to find the factors, if possible. This procedure and the previous two procedures are based on the distributive property.
We will use the distributive property in reverse.
We notice that in the product, is common to both terms. (In fact, is a common factor of both terms.) Since is common to both terms, we will factor it out and write
Now we need to determine what to place inside the parentheses. This is the procedure of the previous section. Divide each term of the product by the known factor
Thus, and are the required terms of the other factor. Hence,
When factoring a monomial from a polynomial, we seek out factors that are not only common to each term of the polynomial, but factors that have these properties:
A monomial factor that meets the above two requirements is called the greatest common factor of the polynomial.
Factor
The greatest common factor is 3.
Factor
Notice that
is the greatest common factor.
Factor
Notice that
is the greatest common factor. Factor out
Mentally divide into each term of the product and place the resulting quotients inside the
Factor
We see that the greatest common factor is
Mentally dividing into each term of the product, we get
Factor
Factor
Factor
Factor
Consider this problem: factor Surely, We know from the very beginning of our study of algebra that letters represent single quantities. We also know that a quantity occurring within a set of parentheses is to be considered as a single quantity. Suppose that the letter is representing the quantity Then we have
When we observe the expression
we notice that is common to both terms. Since it is common, we factor it out.
As usual, we determine what to place inside the parentheses by dividing each term of the product by
Thus, we get
This is a forerunner of the factoring that will be done in Section
Factor
Notice that
is the greatest common factor. Factor out
Factor
.
Notice that
and
are common to both terms. Factor them out. We’ll perform this factorization by letting
Then we have
Factor
Factor
For the following problems, factor the polynomials.
( [link] ) A quantity plus more of that quantity is What is the original quantity?
( [link] ) Solve the equation
( [link] ) Given that is a factor of find the other factor.
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