Consider the polynomial P(x), shown in standard form and in factored form.
p(x)= -(1)/(5)x^(3)+(1)/(5)x^(2)+(17)/(5)x+3= -1/5(x+3)(x+1)(x-5)
(a) State the behavior at the ends (fill in blanks):
At the left, as x „_ ¡V„V, P(x) „_ __ (choose ¡V„V or „V). At the right, as x „_ „V, P(x) „_ __ (choose ¡V„V or „V).
(b) State the y-intercept:
(c) State the x-intercepts:
See:
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(a)
At the left, as x approaches negative infinity, P(x) approaches negative infinity (P(x) → -∞).
At the right, as x approaches positive infinity, P(x) approaches positive infinity (P(x) → +∞).
(b)
The y-intercept can be found by setting x = 0 in the polynomial.
P(0) = -(1/5)(0+3)(0+1)(0-5) = -(1/5)(3)(1)(-5) = 3
So, the y-intercept is 3.
(c)
The x-intercepts can be found by setting P(x) = 0, i.e. -(1/5)(x+3)(x+1)(x-5) = 0.
This equation can be satisfied when x+3 = 0, x+1 = 0, or x-5 = 0.
Solving these equations, we get x = -3, x = -1, and x = 5.
So, the x-intercepts are -3, -1, and 5.
(a) To determine the behavior at the ends, we need to look at the leading term of the polynomial, which is the term with the highest exponent. In this case, the leading term is -(1/5)x^3.
1. At the left end, as x approaches negative infinity (x -> -∞), the leading term -(1/5)x^3 becomes increasingly negative. So, P(x) approaches negative infinity as x approaches negative infinity. We can state this as P(x) approaches -∞ as x -> -∞.
2. At the right end, as x approaches positive infinity (x -> ∞), the leading term -(1/5)x^3 becomes increasingly positive. So, P(x) approaches positive infinity as x approaches positive infinity. We can state this as P(x) approaches +∞ as x -> ∞.
(b) To find the y-intercept, we set x = 0 and evaluate P(x).
P(x) = -(1/5)x^3 + (1/5)x^2 + (17/5)x + 3.
Setting x = 0, we get:
P(0) = -(1/5)(0)^3 + (1/5)(0)^2 + (17/5)(0) + 3
P(0) = 3.
Therefore, the y-intercept is 3. We can state this as the point (0, 3).
(c) To find the x-intercepts, we set P(x) = 0 and solve for x.
From the factored form of P(x):
P(x) = -1/5(x + 3)(x + 1)(x - 5).
Setting P(x) = 0, we have:
0 = -1/5(x + 3)(x + 1)(x - 5).
To find the x-intercepts, we set each factor equal to zero:
x + 3 = 0, x + 1 = 0, and x - 5 = 0.
Solving each equation:
x + 3 = 0 -> x = -3
x + 1 = 0 -> x = -1
x - 5 = 0 -> x = 5
Therefore, the x-intercepts are -3, -1, and 5. We can state this as the points (-3, 0), (-1, 0), and (5, 0).