Thanks for the lovely lesson. Are we writing answers for your book?
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 , it is a perfect square trinomial and has equal factors.
Example 1:
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:
Example 3:
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:
If this trinomial is to be factored into two binomials, the first terms will each be x. 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.
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.
Checking each product, we find the combination of correct factors.
Example 5:
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 )(x) or (2x)(5x). Of course, the remaining steps will take more effort because of the numerous possibilities.
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
Finding monomial factors first can save you much trouble and assure you of results that are factored completely.
Example 6:
Factoring out the monomial here presents the same problem as in the example above.
When factoring, always look 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