Factoring with a Graph, and Synthetic Division:
WHAT DID YOU SAY?
1. At the final paragraph( 15), can anyone tell me what this church is saying--exactly?
A GOOD EXAMPLE TO FACTOR:
2. A student taking a course in pre-calculus sent me this example. The problem was to factor completely the polynomial at L1. It is a good example for factoring polynomials because it is not too complicated or too simple.
3. Synthetic division is used in this hub. An outstanding job done in explaining how to do synthetic division can be found at this hub by Samanthamayer(linked). She gives a very detailed description. Another hub that is just as good on this subject is Julie Burke's hub( linked). MKMenatti( linked) has additional valuable information on this subject. Between these 3 hubs you will be set to go on synthetic division. In many instances synthetic division can be faster and more efficient in yielding information about a function than a calculator. When it is used in conjunction with Descartes' rules of signs you can determine information about the zeros( or roots) of a function.
4. Right off the bat I made a mistake at L1. The "f(x)=" was supposed to be in front of that polynomial, not in the corner. As I said at paragraphs 14 and 15 of hub#12.3, I have a thing about wasting paper, and I avoid it if I can.
5. There are ways to determine the zeros( where the graph crosses the x-axis) without a calculator, but it does not make sense to me to do that when a calculator makes the job much easier and faster. I set my windows at X[ --50, 50], and Y[ --10, 10]. I want the X wide enough to include the lower and upper bounds; however, where the range( the Y values) are set is no big deal since we are only interested where the graph crosses the x-axis, which you want centered between the minimum and maximum Y values; that is you want Y = 0 at the center of the screen. Having done that I graphed L1 and found zeros at about -20, -1, and 6. We set the windows again to get a more accurate idea of where the zeros are: X[ -22, 9]. This new representation of the graph gives zeros at about -20, -1.22, and 7.
USE SYNTHETIC DIVISION TO CHECK FOR ZEROS:
The 3 links at paragraph 3 describe how to do synthetic division, which is used at L2, including L2.1. The last number at L2.1 is zero, which means that X = -20 is a zero, or root of the polynomial at L1. The reason for this is that the upper left corner where the -20 is at L2 can be thought of as f(x), the functional notation for Y. So what that corner says is f(-20) = 0, or Y = 0 at x = -20. But if Y = 0, then this is the x-axis; therefore, this is a root of L1 at x = -20. If you plug in a -20 into the polynomial at L1, you will get an answer of zero. Whatever number is put into that corner will give you the answer of f( x = whatever number) to the polynomial at L1. This will be explained further when we get to L15, and L16.
L4 THROUGH L6 EXPLAINED:
7. Remember to go to paragraph 3 where the 3 links will describe synthetic division( SD), if you are not familiar with it.
8. The 3rd line in SD will give another polynomial of one less degree than that of the dividend( L1 in this case). So the 3rd line of L2, and L2.1 represents the coefficients of the polynomial of L4. Notice it is one degree less than L1. This is a factor of L1; the other factor being (X + 20). Since L4 is a factor of L1, we can try 7( mentioned at the last line of paragraph 5) to see if it is a root of L4, and L4.1. Using SD again at L5 we do get a zero for the last number in the 3rd line; therefore, x = 7 is another root of L1. The last line of L5 represents the coefficients of the polynomial at L6. So now we have 3 factors for L1, which are (X + 20), (X -- 7) and L6.
THE RATIONAL ZEROS THEOREM:
9. The rational zeros theorem states that if the polynomial P(x) has integer coefficients, then every rational zero of P(x) is of the form, p/q, where p is a factor of the constant coefficient, and q is a factor of the leading coefficient. If the root is rational, then it will have this form. The factors for p--which is -404 at L6--are 2,2, and 101. A + and - is put in front of each one of those, as for q also. The factors for q, which is -12 are 2,2 and 3. What we are hoping for is a zero( root) that is rational and close to --1.22. So for p/q we have 2,2,101 over 2,2,3, with +,- in front of each one. 101 is out of the game--it is too big. The highest number we can get to with the two 2's taken one at a time on top is one; therefore, we need them both to get to --1.22. So we know the value of p is 4. But 4 divided by what? The only number that comes closest to --1.22 is the three; therefore, our p/q is 4/3. We have + or - in front of each factor; therefore, we make one of them a minus so we have a minus 4/3.
L7 THROUGH L9 EXPLAINED:
10. Synthetic division is used to check if --4 / 3 is a zero, and since the last number of L7( at the 3rd line) is zero , then --4 \ 3 is a zero of L6, and of course a zero of L1 also. The zeros we have so far are listed at L8. At L9 we have those factors plus the quadratic factor that still needs to be factored, but before we do that, let's take a look at the 3 factors preceding it. OK, f( x) = Y, and Y equals the polynomial at L1. To find the zeros of a polynomial, we equate Y = f( x) = 0. Then we set out to find the value of x that will make f( x = whatever value) = Y = 0. We solve for x. L9 is equated to zero, and if we divide both sides of that equation by each factor then we can isolate any factor we want because zero divided by any number is zero; therefore, we do this with the last three factors of L9 and what we have left is (x + 20) = 0. For this to be true, x must equal a --20, and if we subtract 20 from both sides of the equation we are left with x = --20. This is why we write the factors at L9 with opposite signs. Our zero is --20, so we write the factor as ( x + 20). The same reasoning applies to the next two factors, (x + 4/3), and (x - 7). If you need help in solving equation as these then academysigma has a good hub on this( linked).
L10 THROUGH L14.2 EXPLAINED:
11. The only thing left to do is factor the quadratic at L9. Finding the roots of a quadratic equation is explained by Cristina327( linked) and Hanibal Smith( an exhaustive presentation(linked)). So the quadratic formula is used at L10, and we get an answer that is a complex number and its conjugate at L12. The enormous power of complex numbers is demonstrated at Complex logarithms( linked). Notice at L13 if we multiply the last two factors we will get x2 not --12x2 ; therefore, we have to include the leading coefficient, --12, as one of the factors. This is because when we wrote out the factors with x minus the complex number and its conjugate, we did not include the --12. This will be the case with the leading coefficient of a polynomial when you are factoring them. When you do the factors at L13, remember it is x minus the entire complex numbers. So it is x minus a negative 1 / 2 in both cases, which is why they are both positive. Then it is a +5i on one and a --5i on the other. I made a MISTAKE at L13 and forgot to put +5i on the second one but it was corrected at L14.1 L14 through L14.1 is the complete factorization of L1.
FINDING UPPER AND LOWER BOUNDS:
12. Let us suppose you do not have a calculator. How would you get an idea on where are the upper bound( where the graph no longer crosses the x-axis going to the right on the x-axis), and the lower bound( where the graph no longer crosses the x-axis going to the left on the x-axis)? A "theorem on bounds for real zeros of polynomials" can be found in many precal textbooks. From Swokowski and Cole's outstanding text, "Precalculus, functions and graphs", 8th edition, we have on page 207:
"Suppose that f(x) is a polynomial with real coefficients and a positive leading coefficient and that f(x) is divided synthetically by x -- c.
(1) If c > 0 and if all numbers in the third row of the division process are either positive or zero, then c is an upper bound for the real zeros of f(x).
(2) If c < 0 and if the numbers in the third row of the division process are alternately positive and negative( and a 0 in the third row is considered to be either positive or negative), then c is a lower bound for the real zeros of f(x)."
13. The first thing to notice about this theorem is it says, "a positive leading coefficient", but L1 has a --12, not a +12; therefore, we multiply all of L1 by a negative 1, and then perform the division at L15 and L15.1. Notice we have alternating signs; therefore, according to the theorem, a --21 is a lower bound. Also according to the theorem we have all positive at the third row of of L16 and L17; therefore, 8 is an upper bound of L1.
LEAST UPPER BOUND, AND GREATEST LOWER BOUND:
14. Numbers going left on the real axis( the x-axis) are lower than the ones on the right; therefore, a minus 20( the last real root going left) is the greatest lower bound. 7( the last real root going right) is the least upper bound. For the above theorem to work you want to go a little lower than a minus 20, and a little higher than 7 as L18.1 mentions. L18 is a reminder that the leading coefficient is to be positive to employ the theorem.
EXACTLY WHAT IS THIS CHURCH SAYING?
15. A couple of good friends brought my attention to a church they discovered. I intend to write a hub about the experience, or at least include it at the end of some of my math hubs. I do not want to influence your opinion at all; therefore, I'll say nothing further about the experience. In your opinion, exactly what do you think this church( The United Church of God) is saying( from their booklet, "Fundamental Beliefs of the United Church of God")?
"Just as human children are the same kind beings as their parents and older siblings, human beings, so will we be the same kind of beings as God the Father and Jesus Christ--divine beings. . . . He nevertheless understood that we will be what They are.
"In fact, God was even more explicit about our destiny in Psalm 82:6, stating His intention for people as, 'You are gods, and all of you are children of the Most High.' Jesus actually quoted from this verse( see John 10:30-36). The truth is that our destiny is to bear the name of the God family( Ephesians 3:14-15). Presently, the one God--that is, the one God family--consists of two diving beings: God the Father and Jesus Christ. But ultimately, God intends to expand this divine family into billions.
" . . . God said He would make man in His own image and likeness( again, verses 26-27). The clear implication is that man was created according to the 'God kind,' so to speak, with God intending to reproduce Himself through human beings.
"As first created, the 'likeness' to God in man is quite restricted--limited to such areas as general resemblance in form, feelings, thought, creative abilities and the capacity to govern--all in a rather inferior sense as compared to God. However, God intends for man to ultimately come to share His divine glory, power, intelligence wisdom and righteous, loving character.
"Jesus is called 'the firstborn among many brethren'( Romans 8:29). Human beings have the wonderful potential to enter the God family and be transformed into the same kind beings the Father and Christ now are( Romans 8:14, 19; John 1:12; 1 John 3:1-2)."