Showing posts with label polynomial. Show all posts
Showing posts with label polynomial. Show all posts

Tuesday, October 22, 2013

Synthetic Division

When dividing polynomials of the form p(x)/(x-a) we can use synthetic division as a shortcut for polynomial long division. Below we divide using traditional polynomial long division and synthetic division side by side.

Divide.
 
Answer: 
Both processes give the same result, x^2 - 3x - 2. However, synthetic division uses only the coefficients and requires much less writing.  To understand synthetic division, we walk you through the process below. Be sure the polynomials are in standard form, that is, each term is arranged in descending order from highest power to lowest.

Step 1: Write the root a determined from (x-a) and the coefficients of the polynomial in the first line.
Step 2: Bring down the first coefficient and we are ready to begin.
Step 3: Multiply a by the first coefficient and write the result under the second coefficient.
Step 4: Add the second column and write the result below.
Step 5: Repeat the process for all of the remaining columns.
Step 6: The numbers along the bottom are the coefficients of the result in standard form beginning with a term of degree one less than the original polynomial. The last number is the remainder.
To finish, clearly present the answer to your reader.  Next we do an example with a remainder. Just as we do in polynomial long division, we add a term that is the remainder over the divisor.

Divide.
Here the root (or zero) of (x+5)  is -5.
Multiplying both sides by the divisor (x+5) we have the following.
We mentioned that the polynomials are required to be in standard form. Sometimes there will be "missing terms."  That is, not all powers will have nonzero coefficients.  In this case, we use 0 as placeholders when performing synthetic division.

Divide.

Answer:
Sometimes the root will be a fraction.

Divide.
Answer:

It is interesting to note that the result has a GCF of 2 and we can do the following algebraic manipulations:
In short, this gives us a method of factoring a more complicated polynomial.
 

YouTube videos:

Video Lecture: Polynomial Division: Synthetic Division (10 minutes from Mathispower4u)

Divide a Trinomial by a Binomial using Synthetic Division

Divide a Polynomial by a Binomial Using Synthetic Division

Divide  Polynomial by a Binomial Using Synthetic Division (with placeholder)
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Sunday, April 21, 2013

Elementary Algebra Exam #3

Click on the 10 question exam covering topics in chapters 5 and 6. Give yourself one hour to try all of the problems and then come back and check your answers.



10. The length of a rectangle is 4 centimeters less than twice its width. The area is 96 square centimeters. Find the length and width. (Set up an algebraic equation then solve it)

 
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Sunday, November 4, 2012

Factoring Trinomials of the Form x^2 + bx + c

In this section we will factor trinomials - polynomials with three terms. Students find this difficult at first. However, with much practice factoring trinomials becomes routine. If a trinomial factors, then it will factor into the product of two binomials.

Factor:
   Step 1: Factor the first term: x^2 = x*x.
   Step 2: Factor the last term. Choose factors that add or subtract to obtain the middle term.
   Step 3: Determine the signs by adding or subtracting the product of the inner and outer terms.
   Step 4: Check by multiplying.
Rather than trying all possible combinations of the factors that make up the last term spend some time looking at the factors before starting step two. Look for combinations that will produce the middle term. Here is the thought process in choosing 3 and 4 in step two above:
   "Can I add or subtract 1 and 12 to obtain 7?" – NO     
   "Can I add or subtract 2 and 6 to obtain 7?" – NO    
   "Can I add or subtract 3 and 4 to obtain 7?" – YES, because +3 + 4 = +7

Factor the trinomials.
This process used for factoring trinomials is sometimes called guess and check or trial and error. The biggest problem occurs when the signs are improperly chosen. With this in mind, you should take care to check your results by multiplying. Also, since multiplication is commutative order does not matter, in other words
If the trinomial has a GCF you should factor that out first.  Also, you should factor in such a way as to ensure a resulting trinomial with a positive leading coefficient.

Factor the trinomials.
Take care to perform the check. Most of the problems that you will encounter factor nicely but be sure to watch out for something like this . The middle term works but the last term does not:
because the sign of the last term is incorrect

Video Examples on YouTube: