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Question 637123: Please help me solve this problem: (x^3 - 13x^2 - 41x + 4) / (x - 7)
Right now I have the answer as x^2 -6x + 48
I don't think what I have so far is right, and I don't know how to finish it. Please help, thank you.
Found 2 solutions by ewatrrr, DrBeeee:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website!
Hi,
(x^3 - 13x^2 - 41x + 4) / (x - 7)
may use Synthetic Divsion for this ex.
7 1 -13 -41 4
7 -42 -581
1 -6 -83 -577
Yields x^2 - 6x - 83 R = -577
Answer by DrBeeee(684)  (Show Source):
You can put this solution on YOUR website! Your suspection that your quadratic is wrong couldn't be truer. The product of the two constant terms, -7 and 48 is not equal to 4 as it should be. As to how you proceed to factor the trinomial, you can take solace in the fact that a trinomial (or any odd powered polynomial) has at least one real root. This is because complex roots occur in pairs or an even number.
Since we have at least one real root, the function must cross the x-axis at least once. I use the trial/average technique to find the real value of x that makes the trinomial equal to zero. That is, we find out where the given trinomial crosses the x-axis. Let's do it, OK?
Let f(x) = x^3 - 13*x^2 - 41*x + 4.
I start with x = 0
Then f(0) = 0 - 13*0 - 41*0 + 4 = 4 > 0
Now try x = 1
f(1) = 1 - 13 - 41 + 4 = -49 < 0
Now we know that f(x) crossed the x-axis somewhere between 0 and 1 (because f(0) is + and f(1) is negative). Now calculate the average of 0 and 1. This is .5, so try x = .5
f(.5) = 0.125 - 3.25 - 20.5 + 4 = - 19.625 < 0.
Now 0 < root < .5, so try the average or 0.25 etc.
f(0.25) < 0
f(0.125) < 0
f(0.0625) > 0
Root lies between 1/16 < x < 1/8
f(3/32) = +0.043 > 0
f(7/64) = -0.638 < 0
After many tries, I determined that
f(0.09473600) = 0.00000042 which is very close to zero.
The first order factor is
(x - 0.094736)
FOIL this with (x^2 + b*x + c) and set the product equal to the given trinomial. Then solve for b and c by equating the coefficients of x^2, x^1 and x^0. This evaluation will give
b = -12.905264 and
c = -42.2226
The factorization is
(x - 0.094736)(x^2 -12.905264x -42.2226) = x^3 -13x^2 - 41x +4
The quadratic can be factored by using the quadratic equation. To help you out, here's my version
x = (-b/2 +/-sqrt((b/2)^2-a*c))/a
where a = 1, b = -12.905264 and c = -42.2226. Plug and grind gives
x = 6.452632 +/- 9.15746 or
x = -2.704828, 15.610092 with factors
(x + 2.704828)(x - 15.610092) which when FOILed yields
x^2 - 12.90526x -4202226 as given above.
The final factorization is
(x - 0.094736)(x + 2.704828)(x - 15.610092) which equals
x^3 -13x^2 -41x +4
Amen
PS some numbers may not agree exactly because of all the decimals values needed.
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