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Finding the zeroes of an equation means finding the x-values for which the function, in this case f(x), is equal to 0.
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Let us set the function's value to 0 and rewrite the equation.
0 = x³ + 2x² - 23x - 60
Using rational root theorem
Find divisors of the constant term -60
-(1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60)
Brute Force
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Let us find one factor from the divisors we listed.
Starting with -1
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0 = (-1) + 2 + 23 - 60 NOPE
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Trying (-2)
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0 = (-8) + 8 - 184 - 60 NOPE
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Trying (-3)
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0 = (-27) + 18 + 69 - 60 YEP
0 = (-1)³ + 2(-1)² - 23(-1) - 60
0 = (-2)³ + 2(-2)² - 23(-2) - 60
0 = (-3)³ + 2(-3)² - 23(-3) - 60
From our brute force troubleshooting we found that (x + 3) can be factored out.
With that factored out, we will turn now to Ruffini's method in order to divide.
(x³ + 2x² - 23x - 60) / (x + 3)
(x² - x - 20) * (x + 3)
(x² - x - 20) = 0 ?
(x - 5)(x + 4)
(x + 3)(x - 5)(x + 4)
x + 3 = 0
x - 5 = 0
x + 4 = 0
ZEROES
x = -3
x = 5
x = -4
Ruffini's method according to AI
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