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On the Factorization of Various Types of Expressions. Henry Blumberg. PNAS June 1, 1 (6) ; xefejevykohi.tk Henry Blumberg.
Table of contents
- REMOVING COMMON FACTORS
- Factorization by Using Identities
- Always the first step: Look for a GCF
- Factorization by Using Identities | Factorize an Algebraic Expression Easily
Let us limit to the above three ways. Which of the above three ways is accepted as standard form of factorization? The third one. In the third one, 6x 2 is the greatest common divisor or highest common factor of the two algebraic expressions monomials 12x 2 and 18x 3. To find the other factor, divide each term by the H. Those terms which yield a common factor on grouping!
Solution: Group terms with similar literal coefficients and numerical coefficients. Solution: Group p 2 qx and px 2 y. Trinomial Perfect Squares have three monomials, in which two terms are perfect squares and one term is the product of the square roots of the two terms which are perfect squares. Factorization Expressing polynomials as product of other polynomials that cannot be further factorized is called Factorization.
Factors that are numbers are not further factorized. Complete Factorization In complete factorization, the greatest common factor is written as the common factor. The product of two even numbers is even. The product of an odd and an even number is even. The sum of two odd numbers is even. The sum of two even numbers is even.
The sum of an odd and even number is odd. Thus, only an odd and an even number will work. We need not even try combinations like 6 and 4 or 2 and 12, and so on. Here the problem is only slightly different. We must find numbers that multiply to give 24 and at the same time add to give - You should always keep the pattern in mind. The last term is obtained strictly by multiplying, but the middle term comes finally from a sum.
Knowing that the product of two negative numbers is positive, but the sum of two negative numbers is negative, we obtain. We are here faced with a negative number for the third term, and this makes the task slightly more difficult. Since can only be the product of a positive number and a negative number, and since the middle term must come from the sum of these numbers, we must think in terms of a difference.
REMOVING COMMON FACTORS
We must find numbers whose product is 24 and that differ by 5. Furthermore, the larger number must be negative, because when we add a positive and negative number the answer will have the sign of the larger. Keeping all of this in mind, we obtain. The order of factors is insignificant. The following points will help as you factor trinomials: When the sign of the third term is positive, both signs in the factors must be alike-and they must be like the sign of the middle term.
Factorization by Using Identities
When the sign of the last term is negative, the signs in the factors must be unlike-and the sign of the larger must be like the sign of the middle term. In the previous exercise the coefficient of each of the first terms was 1. When the coefficient of the first term is not 1, the problem of factoring is much more complicated because the number of possibilities is greatly increased.
Having done the previous exercise set, you are now ready to try some more challenging trinomials. Notice that there are twelve ways to obtain the first and last terms, but only one has 17x as a middle term. You could, of course, try each of these mentally instead of writing them out. In this example one out of twelve possibilities is correct.
Thus trial and error can be very time-consuming. Even though the method used is one of guessing, it should be "educated guessing" in which we apply all of our knowledge about numbers and exercise a great deal of mental arithmetic. In the preceding example we would immediately dismiss many of the combinations.
Since we are searching for 17x as a middle term, we would not attempt those possibilities that multiply 6 by 6, or 3 by 12, or 6 by 12, and so on, as those products will be larger than Also, since 17 is odd, we know it is the sum of an even number and an odd number. All of these things help reduce the number of possibilities to try. First find numbers that give the correct first and last terms of the trinomial.
Then add the outer and inner product to check for the proper middle term. First we should analyze the problem.
The last term is positive, so two like signs. The middle term is negative, so both signs will be negative. The factors of 6x2 are x, 2x, 3x, 6x. The factors of 15 are 1, 3, 5, Eliminate as too large the product of 15 with 2x, 3x, or 6x.
Try some reasonable combinations. These would automatically give too large a middle term. See how the number of possibilities is cut down. Analyze: The last term is negative, so unlike signs. We must find products that differ by 5 with the larger number negative. We eliminate a product of 4x and 6 as probably too large. Try some combinations. Remember, mentally try the various possible combinations that are reasonable. This is the process of "trial and error" factoring.
You will become more skilled at this process through practice. Be careful not to accept this as the solution, but switch signs so the larger product agrees in sign with the middle term.
Always the first step: Look for a GCF
By the time you finish the following exercise set you should feel much more comfortable about factoring a trinomial. Upon completing this section you should be able to: Identify and factor the differences of two perfect squares. Identify and factor a perfect square trinomial. In this section we wish to examine some special cases of factoring that occur often in problems. If these special cases are recognized, the factoring is then greatly simplified.
Recall that in multiplying two binomials by the pattern, the middle term comes from the sum of two products. From our experience with numbers we know that the sum of two numbers is zero only if the two numbers are negatives of each other.
Factorization by Using Identities | Factorize an Algebraic Expression Easily
When the sum of two numbers is zero, one of the numbers is said to be the additive inverse of the other. In each example the middle term is zero. This is the form you will find most helpful in factoring. Special cases do make factoring easier, but be certain to recognize that a special case is just that-very special. In this case both terms must be perfect squares and the sign must be negative, hence "the difference of two perfect squares. The sum of two squares is not factorable. You must also be careful to recognize perfect squares. Remember that perfect square numbers are numbers that have square roots that are integers.
Also, perfect square exponents are even. Students often overlook the fact that 1 is a perfect square. Thus, an expression such as x 2 - 1 is the difference of two perfect squares and can be factored by this method. Another special case in factoring is the perfect square trinomial. Observe that squaring a binomial gives rise to this case. We recognize this case by noting the special features. Three things are evident. The first term is a perfect square. The third term is a perfect square. The middle term is twice the product of the square root of the first and third terms.
For factoring purposes it is more helpful to write the statement as. To factor a perfect square trinomial form a binomial with the square root of the first term, the square root of the last term, and the sign of the middle term, and indicate the square of this binomial. Always square the binomial as a check to make sure the middle term is correct.
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Upon completing this section you should be able to: Find the key number of a trinomial. Use the key number to factor a trinomial. In this section we wish to discuss some shortcuts to trial and error factoring. These are optional for two reasons. First, some might prefer to skip these techniques and simply use the trial and error method; second, these shortcuts are not always practical for large numbers. However, they will increase speed and accuracy for those who master them. The first step in these shortcuts is finding the key number. After you have found the key number it can be used in more than one way.
In a trinomial to be factored the key number is the product of the coefficients of the first and third terms. The product of these two numbers is the "key number. Solution Step 1 Find the key number. Place these factors in the first and last positions in the pattern There is only one way it can be done correctly. Step 5 Forget the key number at this point and look back at the original problem. Since the first and last positions are correctly filled, it is now only necessary to fill the other two positions. Again, this can be done in only one way.
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We know the product of the two first terms must give 4x 2 and 4x is already in place. There is no choice other than x. Note that in step 4 we could have started with the inside product instead of the outside product. We would have obtained the same factors. The most important thing is to have a systematic process for factoring. We have no choice other than - 5. Remember, if a trinomial is factorable, there is only one possible set of factors.
If no factors of the key number can be found whose sum is the coefficient of the middle terms, then the trinomial is prime and does not factor.
A second use for the key number as a shortcut involves factoring by grouping. It works as in example 5.