Question 192853: Please help me to solve this polynomial:
f(x)=(x-8)(x+3)(x-9)(x+8)(x-2)(x+5)(x-2)
I have to write ALL the process down. I tried it and got a result, but when I graph it, it was different to my original function.
Thanks!
Answer by RAY100(1637)   (Show Source):
You can put this solution on YOUR website! This is an interesting problem, if only from its size.
we started by arranging the factors in order.
y=(x-9)(x-8)(x-2)(x-2)(x-3)(x-5)(x-8)
note this is a 7th degree function. This means(7 roots) and it is an odd function
we can find the roots from the factors, for example set x-9=0, x=-9 root likewise the roots in order are, 9,8,2,2,3,5,8.
Note that the root (2) happens twice, this means the function curve just touches the y=0 axis but does not pass thru. With the other roots the function intersects and passes thru.
note it is a positive function. If we look close the leading coefficient will be plus, because we have no negative (x) in factors. After we multiply all factors, this holds true.
Summarizing, it is a positive odd function, with 7 roots, and these roots are defined by the factors.
The odd positive nature indicates as we graph the function starts at the left with a negative and moves upward to positive right.
If we use a standard x - y coordinate system and make just a rough sketch, we can define the shape of the function.
starting at lower left we rise thru x=(-8), turn and pass downward thru x=(-5) turn and pass upward thru (x=-3), turn downward and just touch the axis at x=(+2) before turning upward and then downward thru x=(8),and then turn and pass upward thru x=(9) and continuing upwards.
We see 6 turns which is consistent with a 7 degree function
We would probably say the problem is complete here, but you seemed to want to expand the function. This is a lot of work and many chances for numerical error.
I started with a clean sheet of paper sideways.
i did 2 factor by themselves, as this allowed the use of foil
1)(x+8)(x-8)= x^2-64
2)(x-2)(x-2)=x^2-4x-4
3)(x+3)(x-7)=x^2 -4x-21
4)(x-5) had no mate
now we multiplied 1*2 and 3*4
5) x^4-4x^3-60x^2+256x-256
6)x^3-x^2-57x-135
and lastly we multiplied 5*6
but we set each distribution on a horizontal line and matched the next distribution below.
This resulted in a vertical accounting of the for factors, one vertical column for each degree.
The result was
y=x^7-5x^6-113x^5+409x^4+3960x^3+9303x^2-19968x+34560
my graphing calc did not like this but i feel the above sketch is correct.
to recheck, i would wait a day and redo
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