which table describes the behavior of the graph of ( f(x)=2x^{3}-26x - 24 )?
which table describes the behavior of the graph of ( f(x)=2x^{3}-26x - 24 )?
Answer
Explanation:
Step1: Find the roots of the function
Set (f(x)=2x^{3}-26x - 24 = 0). Factor out (2): (2(x^{3}-13x - 12)=0). By trial - and - error (using the Rational Root Theorem, where possible rational roots are factors of (12) divided by factors of (1)), we find that (x=-1) is a root. Using polynomial long - division or synthetic division: ((x^{3}-13x - 12)\div(x + 1)=x^{2}-x - 12). Factor (x^{2}-x - 12=(x + 3)(x - 4)). So the roots of (f(x)) are (x=-3,x=-1,x = 4).
Step2: Test intervals
- For the interval ((-\infty,-3)), let (x=-4). Then (f(-4)=2\times(-4)^{3}-26\times(-4)-24=2\times(-64)+104 - 24=-128 + 104-24=-48<0) (below the (x) - axis).
- For the interval ((-3,-1)), let (x=-2). Then (f(-2)=2\times(-2)^{3}-26\times(-2)-24=2\times(-8)+52 - 24=-16 + 52-24 = 12>0) (above the (x) - axis).
- For the interval ((-1,4)), let (x = 0). Then (f(0)=2\times0^{3}-26\times0-24=-24<0) (below the (x) - axis).
- For the interval ((4,\infty)), let (x = 5). Then (f(5)=2\times5^{3}-26\times5-24=2\times125-130 - 24=250-130 - 24 = 96>0) (above the (x) - axis).
Answer:
The table with intervals ((-\infty,-3)) (below), ((-3,-1)) (above), ((-1,4)) (below), ((4,\infty)) (above)