This week, we will cover the topics of factoring polynomials and simplifying rational expressions.
There is an easy technique for factoring integers using a calculator. The calculator does not even need to be in scientific mode. Take a list of the prime integers and divide each one into the number being factored. If the number is evenly divisible (nothing to the right of the decimal point) then the divisor is a factor. Use the quotient as the new number to factor and continue dividing, starting with the last factor in case it occurred more than once. Let's factor the popular ZIP code, 90210. Starting from the bottom, divide by the first prime number, 2. The quotient, 45105, is odd, so we try the second prime number, 3. If we try to divide 15035 by 3 again, there is a remainder. The next prime number, 5, divides evenly into the 15035, giving the quotient of 3007. None of the subsequent prime numbers, (7, 11, 13, 17, 19 ...) divide evenly into 3007 until 31. The quotient, 97 is a prime.
97
____
31)3007
_____
5)15035
_____
3)45105
_____
2)90210So the prime factors of 90210 are 2, 3, 5, 31, and 97.
As I mentioned in the first week's lecture, simplifying a polynomial can often increase the speed of a calculation. The procedure for factoring a polynomial tends to be a bit more complicated, since there is no orderly list of numbers to divide by. It is necessary to examine the polynomial to find a divisor.
In order to identify the most likely factors, the
first step is to arrange the terms in the order of the exponents. If
there are multiple variables, select the one with the highest
exponent to sort by. For example, the sequence of
66t3v2
+ 55t4v2 + 88 t2v2 +
22t5v3 would be 22t5v3 +
55t4v2 + 66t3v2 +
88 t2v2 , since the variable “t”
has the highest exponent.
The next step is to split up the coefficients to prime factors. In the example, the result would be (2*11)t5v3 + (5*11)t4v2 + (2*3*11)t3v2 + (2*2*2*11) t2v2 . This shows whether there are any factors common to each of the terms so that we can separate these monomial factors. In our example, the common factors are 11, v2, and t2. Separating these factors gives (11 v2 t2 )(2t3v + 5t2 + 6t + 8).
The standard format for a trinomial is ax2 + bx + c, where a, b, and c are coefficients. The “ac” method of factoring involves finding two numbers that add up to b and have the product ac. So the first step in factoring is to separate a and c into prime factors. As an example, try warm-up 4 on page 299 of the textbook.
To factor 15 x2 + 31 x + 14, note that 15 = 3*5 and 14 = 2*7; the factors of 210 are 2,3,5,7. We can arrange the possible factors into a list
|
First |
Second |
Sum |
|
1 |
210 |
211 |
|
2 |
105 |
107 |
|
3 |
70 |
73 |
|
5 |
42 |
47 |
|
7 |
30 |
37 |
|
6 |
35 |
41 |
|
10 |
21 |
31 ß |
|
14 |
15 |
29 |
Only 10 and 21 have a sum of 31 so replace the
middle term of (15 x2 + 31 x + 14) by “21x +
10x” to yield
(15 x2 + 21 x + 10x + 14) =
3x(5x + 7) + 2(5x + 7) = (3x + 2) (5x + 7).
With four factors, how can we be sure that every possible combination is listed? One way is to group by the number of prime factors in the first multiplier, (a) no prime factor, (b) one prime factor, (c) two prime factors. We do not consider four prime factors because they will be in the second multiplier when the first multiplier has none. In the same way, we don’t have to consider three prime factors.
There are only four choices for picking one of the available prime factors, (2, 3, 5, and 7).
To pick all of the choices for a pair of prime factors, first consider those with the first prime factor included. There are only three of these (2,3), (2,5), (2,7).
Now, consider those without the first prime factor. Picking two factors from a list of three (3, 5, and 7) is the same as excluding one of them, so the choices are (5,7), (3,7), and (3,5). But these are the items in the second multiplier when we considered the first three pairs of factors on our list.
This method of listing all combinations is a preview of finite mathematics.
With a larger polynomial, the strategy on page 304 of the textbook is helpful. Using the trial and error method can be useful, but time consuming. The choices for divisors will not be overwhelming, because the coefficients for the trial divisor (ax + b) will be factors of the first and last coefficients of the polynomial, respectively.
Rational expressions are those involving division.
Numerator
Denominator
Multiplying the numerator and the denominator of the expression by the same number does not change the value (unless the multiplier is zero), so the same fraction can be expressed in an infinite number of ways. In most cases, the rational expression should be simplified, so that the numerator and denominator have no common factors; one exception to this rule is expressing ten minutes as 10/60 hours.
The standard procedure for reducing a fraction is to split the numerator and denominator into prime factors and cross out those that match in both.
120 = 2*2*2*2*3*5 = 2*2*2*2*3*5 = 2
60 2*2*2*3*5 2*2*2*3*5
Instead of crossing out n occurrences of a factor, we can subtract n from the exponent.
120 = 24*3*5 = 24-3*3*5 = 260 23*3*5 23-3*3*5
There are exclusions to avoid hidden division by zero. You cannot divide both the denominator and the numerator by (x–8) if x=8.
To multiply two rational expressions, multiply the numerators and then multiply the denominators, in other words, place them side-by-side and extend the line between them.
3 × 2 = 3 * 2 = 2 * 3 = 1 4 3 2 * 2 * 3 2 * 2 * 3 2
To divide two rational expressions, multiply the dividend by themultiplicative inverse of the divisor. In other words, turn the number you're deviding by upside down and then multiply.
3 ÷ 3 = 3 × 2 = 3 * 2 = 2 * 3 = 1 4 2 4 3 2 * 2 * 3 2 * 2 * 3 2
One of the techniques of manipulating figures is to keep the units of measure with the numbers. To convert 20 miles per hour to feet per second:
20 miles * 5280 feet * 1 hour = 20 miles * 5280 feet * 1 hour = 29.33 feet 1 hour 1 mile 3600 seconds 1 hour 1 mile 3600 seconds second
Working with units of measure can prevent many careless errors. It is used in physics and chemistry for a “sanity check”.
To add two rational expressions, convert both expressions so that they have the same denominator, preferably the lowest common denominator (LCD). Note that the denominator does not have to be the lowest as long as it is common to both expressions being added. When you add a quarter and a dime, (¼) dollar + (1/10) dollar, you usually add 25/100 + 10/100 to get 35 cents. You do not add (4/20) + (2/20) to get 7/20 of a dollar; but if you wanted to, multiply the quarter by 1 in the form of 5/5 and multiply the dime by one in the form of 2/2.
1 * 5 + 1 * 2 = 5 + 2 = 7
4 * 5 10 2 20 20 20
To get the lowest common denominator of two expressions, first find the prime factors of both denominators. Then multiply top and bottom of the first by all factors in the second denominator that are not in the first denominator; do the same for the second denominator. When you're finished, both expressions have the same denominator; just add the numerators and simplify the result.
Find the prime factors of your ZIP code, (if you want to keep your address private, pick the ZIP of your favorite magazine).
Which format would you use to express 14 ounces in cups, 7/4 or 1 ¾? Would the application influence your decision?
How would you factor x2 + 19x – 144? How do you arrange the factors?
Can you think of an example of units of measure in calculations being helpful in your work?