Question 63227: Hi, Stan just answered a question for me but I need more explanation. Here was my e-mail and his response. "How do you find an equation for a grph that goes through the x-axis 4 (or more) times? There are no specific points except for (a,0) (b,0) (c,0) and (d,0).
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a,b,c, and d are called "zeroes" of the polynomial
The polynomial has factors (x-a),(x-b),(x-c), and (x-d)
f(x)=k(x-a)(x-b)(x-c)(x-d)
Note: k is some non-zero constant"
I was wondering how you get the factors, what are zeroes, and how did he come up with "k". Thank You!!
Found 2 solutions by stanbon, venugopalramana:
Answer by stanbon(75887)   (Show Source):
Answer by venugopalramana(3286)  (Show Source):
You can put this solution on YOUR website! Hi, Stan just answered a question for me but I need more explanation. Here was my e-mail and his response. "How do you find an equation for a grph that goes through the x-axis 4 (or more) times? There are no specific points except for (a,0) (b,0) (c,0) and (d,0).
------------------------------------------
a,b,c, and d are called "zeroes" of the polynomial
The polynomial has factors (x-a),(x-b),(x-c), and (x-d)
f(x)=k(x-a)(x-b)(x-c)(x-d)
Note: k is some non-zero constant"
I was wondering how you get the factors, what are zeroes, and how did he come up with "k". Thank You!!
LET US CALL THE POLYNOMIAL AS 'Y'FOR EASE OF COMMUNICATION.LET US CONSIDER 2 SIMPLE EXAMPLES
1.DEGREE 1
Y = X+5=0.....SO WE KNOW X=-5..HENCE WE SAY
A)X=-5 IS A ZERO OF THE POLYNOMIAL OR
B)X=-5 IS X INTERCEPT AS THE GRAPH OF Y = X+5 CUTS THE X AXIS AT X=-5
C)WE SAY X+5 IS A FACTOR OF Y THE POLYNOMIAL
2.DEGREE 2
Y =X^2-1=0...SO WE KNOW X=+1...OR ....X =-1 AND IN SAME WAY AS ABOVE, WE SAY
A)X=+1 OR X =-1 ARE ZEROS OF THE POLYNOMIAL OR
B)X=+1 OR X =-1 ARE X INTERCEPTS AS THE GRAPH OF Y = X^2-1 CUTS THE X AXIS AT X=+1 AND X = -1
C)WE SAY X+1 AND X-1 ARE FACTORS OF Y THE POLYNOMIAL
SO ALL THE 3 ABOVE USAGES ARE ESSENTIALLY SIMILAR/SAME,THE METHOD OF WORKING
BEING IDENTICAL AND ONLY THE ANSWER IS EXPRESSED IN DIFFERENT WAYS AS EXPLINED ABOVE.
THE METHOD IS WE EQUATE THE POLYNOMIAL TO ZERO AND FIND VALUES OF X WHERE THAT WILL HAPPEN.
NOW IN THE ABOVE PROBLEM YOU WERE ALREADY GIVEN THE GRAPH GOES THROUGH
(A,0),(B,0),(C,0),(D,0)...NOTE Y COOORDINATE IS ZERO EVERY WHERE. THAT IS THE CURVE IS MEETING X AXIS AT A,B,C,D.SINCE ON X AXIS Y COORDINATE = 0.
SWO WE ARE GIVEN THE DATA IN FORM B...THE XINTERCEPTS.
HENCE AUTOMATICALLY A AND C FOLLW..THAT IS
A) X=A OR B OR C OR D ARE THE ZEROS OF THE POLYNOMIAL AND
C)X-A,X-B,X-C,X-D ARE THE FACTORS OF POLYNOMIAL.
HENCE THE POLYNOMIAL IS A PRODUCT OF THESE 4 FACTORS AND A CONSTANT 'K'SAY
SO
Y = K(X-A)(X-B)(X-C)(X-D)
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