Help i dont understand how to do theses problems

ALGEBRA II – UNIT 4

POLYNOMIALS

FACTORING TRINOMIALS

Since factoring is essentially "un-multiplying," we must use our multiplication skills backwards to find factors. If a trinomial is in the form http://media01.owotw.com/mat1107/4/l3_htm3ab.gif, it is a perfect square trinomial and has equal factors.

Example 1:

http://media01.owotw.com/mat1107/4/l3_htm3a.gif

http://media01.owotw.com/mat1107/4/l3_htm3a%20copy.gif

Notice that when you square a binomial that is a sum, the middle term will be positive, and when you square a binomial that is a difference, the middle term will be negative.

If the trinomial is more complicated, you should still recognize it as a perfect square by observing the first and third terms as perfect squares, and noticing that the middle term is twice the product of the square roots. Example 3 shows that you can rearrange terms if necessary.

Example 2:

http://media01.owotw.com/mat1107/4/l3_htm5.gif

Example 3:

http://media01.owotw.com/mat1107/4/l3_htm6.gif

Some trinomials must be factored by using a careful inspection method. In these cases, you must think like a detective and try to "un-foil" the trinomial into a product of two binomials.

Example 4:

http://media01.owotw.com/mat1107/4/l3_htm7.gif

If this trinomial is to be factored into two binomials, the first terms will each bex.The negative sign in -18 tells us that it is a product of a negative and positive number. Write one negative sign and one positive sign.

http://media01.owotw.com/mat1107/4/l3_htm8.gif

Since the sign of the linear term (middle term), -3x, is negative, we can conclude that the factor of -18 with the larger absolute value will be negative. Write the various sets of factors of 18, making the larger one negative.

http://media01.owotw.com/mat1107/4/l3_htm9.gif

Checking each product, we find the combination of correct factors.

http://media01.owotw.com/mat1107/4/l3_htm10.gif

Example 5:

http://media01.owotw.com/mat1107/4/l3_htm11.gif

In this case, more possibilities exist and you will need to try more factors. You can see that now there are the factors of the first term of the trinomial to consider as well as the last term. You may start with (10x http://media01.owotw.com/mat1107/4/invis.gif)(xhttp://media01.owotw.com/mat1107/4/invis.gifhttp://media01.owotw.com/mat1107/4/invis.gif) or (2xhttp://media01.owotw.com/mat1107/4/invis.gifhttp://media01.owotw.com/mat1107/4/invis.gif)(5xhttp://media01.owotw.com/mat1107/4/invis.gifhttp://media01.owotw.com/mat1107/4/invis.gif). Of course, the remaining steps will take more effort because of the numerous possibilities.

http://media01.owotw.com/mat1107/4/l3_htm13.gif

The real challenge in factoring trinomials is organizing the possibilities and systematically trying them until you find the right ones. This process takes patience.

Below is a list of all the possible binomial factors you might have to check when trying to factor the preceding example.

(x - y)(10x - 12y)

(x - 3y)(10x - 4y)

(x - 2y)(10x - 6y)

(2x - y)(5x - 12y)

(2x - 3y)(5x - 4y)

(2x - 2y)(5x - 6y)

Notice that there are two factors of the first term and three factors of the second term. The result is six possible binomial factors for the trinomial. However, only one pair will give the right middle term when you multiply!

Factoring Out the Greatest Common Monomial Factor

http://media01.owotw.com/mat1107/4/lin2%21409.gif

Finding monomial factors first can save you much trouble and assure you of results that are factored completely.

Example 6:

http://media01.owotw.com/mat1107/4/l3_htm14.gif

Factoring out the monomial here presents the same problem as in the example above.

When factoring,alwayslook first for a monomial factor. Doing this may greatly simplify the problem.

Factor completely, then place the factors in the proper location on the grid.

x2 - 8x + 16

Factor completely, then place the factors in the proper location on the grid.

c2 + 6c + 9

Factor completely, then place the factors in the proper location on the grid.

16x2 +48xy + 36y2

Factor completely, then place the factors in the proper location on the grid.

25a2 - 70a + 49

Factor completely, then place the factors in the proper location on the grid.

16ax + 4x2 +16a2