4.1+4.8 Polynomial Behavior
Identify if the following function is a polynomial.
f(x) = -3x2 + 7x5 - 1/x + 2x
If YES, write it in standard form.
What is ... no?
Simplify to find the sum:
(-12x2 + 6x3 - 7) + (10 - x2 + 2x3 + 4x)
8x3 - 13x2 + 4x + 3
Factor the polynomial completely.
j(x) = -96x6 - 16x4 + 4x2
j(x) = -4x2 (24x4 + 4x2 - 1)
*always take out negative from first term if you can
Where are the asymptotes and holes for the rational function:
((x - 6) (x + 7))/((x+7)(3x+6)
2. Vertical Asymptote when x = -2
3. Horizontal Asymptote when y = 1/3
The degree, type, and leading coefficient of the polynomial:
g(x) = -8 + 6x2 - 3x + 0.5x3
Degree = 3
Type = Cubic
Leading Coefficient = 0.5
Find the product.
(2x2 + 5x - 3)(4x - 2)
8x3 + 16x2 - 22x + 6
Factor the polynomial completely.
y = 27g6 + 8g3
y = g3(3g + 2) (9g2 - 6g + 4)
*Difference of cubes:
a3 + b3 = (a + b) (a2 - ab + b2)
Match the process with the operation (+, -, x, /)
a. (1)Multiply for common denominator (2)Distribute and combine like terms in the numerator (3)Factor if needed until all exponents=1, if possible
b. (1)Multiply across without FOILing factors (2)Factor if needed until all exponents=1, if possible
c. (1)Keep, change, flip (2)Multiply across without FOILing factors (3)Factor if needed until all exponents=1, if possible
a. Addition and Subtraction + , -
b. Multiplication X
c. Division /
The x-intercepts, local maximums, and local minimums of the graph
h(x) = 0.5x2 + x - 5
x-intercepts = (-4.3, 0) and (2.3, 0)
local minimum = (-1, -5.5)
No local maximums
Find the product.
(5t - 9)2
25t2 - 90t + 81
Factor the polynomial completely.
n(x) = -x5 + 3x4 + 54x3
n(x) = -x3 (x - 9) (x + 6)
Simplify the following expression into factors to identify any holes and asymptotes.
y=(x-1)/(5x^5)*(x^5*(x+4)(x-6))/(x-1)^2
Hole: x = 1,
Vertical asymptote: x = 1,
Horizontal asymptote: y = 1/5
Simplified:
y=((x+4)(x-6))/(5(x-1)(x-1))
Describe the end behavior of the function
f(x) = -2x5 + 8x3 - 9
As x -> -∞ , f(x) -> ∞
As x -> ∞ , f(x) -> -∞
Find the product.
(k - 2)(k + 1)(k + 4)
k3 + 3k2 - 6k - 8
Factor the polynomial completely.
m(x) = 2x3 + 3x2 - 32x - 48
m(x) = (x - 4) (x + 4) (2x + 3)
*Remember: When is a polynomial completely factored?
Simplify into factored form to identify any holes and asymptotes.
y=(3x)/(x-5)-3/(x+2)
y=(3x^2+3x+15)/((x-5)(x+6)
Holes: None (No factors canceled)
Vertical asymptotes: x = 5, x = -6
Horizontal Asymptote: y = 3/1 or 3
Describe the intervals where f(x) = x2 + 12x - 28 is
a) positive, negative
b) increasing, decreasing
a) Positive: (-∞ , -2) and (14, ∞ )
Negative: (-2, 14)
b) Increasing: (6, ∞)
Decreasing: (-∞ , 6)
Find the quotient.
(3x3 + 4x2 + x - 2) / (x - 1)
3x2 + 7x + 8 + 6/(x-1)
Factor and solve the polynomial.
-b3 + 10b2 - 21b = 0
Factored: y = -b (b - 7) (b - 3)
Solutions: b = 0, b = 7, b = 3
Solve the following equation.
g(x)=(x+3)/(x-3)+x/(x-5)=(x+5)/(x-5)
x = 0 and x = 7