Algebra is the study of using symbols to ex press
mathematical ideas. The topic in this
and the next lesson is factoring polynomials .
Factoring
There are many situations where it is necessary to write a polynomial in
factored form to
solve a problem . For example, to reduce a fraction it is necessary to consider
factors. If
you were asked to reduce the fraction 15 / 20, you would factor 5 out of the
numerator
and denominator and then cancel that factor. 15 / 20 = 3(5) / 4(5) = 3/4.
If the average cost of producing x units of a product is expressed as
and we wish to know how many units to produce to keep the average cost at or
be low $400,
we will want to write the polynomial in the numerator in factored form (x + 8)(x
+ 5).
(x + 8)(x + 5)/(x + 8) = x + 5 = 400. We can produce 395 units and keep the
average cost
at 400.
Recall from lessons 10 and 11 that we multiply like
variables by adding exponents .
Examples
To begin factoring a polynomial we wish to factor out the
largest monomial possible. For
each type of factor, how many are in common to each term?
Examples
Notice that you can check the accuracy of factoring by
multiplying the factors out. You
will obtain the original polynomial. In the third example above you could check
by
multiplying
Factoring Completely
When you partially factor a polynomial it is not considered completely factored.
For
example if you factored 3x^4 - 6x^2 by pulling out x, it would not be complete
because the
second factor could be factored further.
Notice how much simpler this problem was when the largest
possible monomial, 3x^2, was
factored out in the first step in the example on the previous page. The same
answer is
obtained but it is simpler to start by factoring out the largest possible
monomial. In any
factoring problem, it is always simpler to firstly factor out the largest
possible monomial
common to each term of the polynomial .
Summary
In any factoring problem, always begin by factoring out the largest monomial
common to
each term.
Example
Factoring trinomials is the topic of the next lesson. For
now, stopping at
2(x2 + 13x + 40) would be acceptable.
Worksheet
