Analyzing Polynomial Functions and Their Properties

End behavior of f(x) = -2x^4 + 7x^2 - x + 9

Falls left and falls right (even degree, negative leading coefficient)

Zeros of f(x) = 9(x - 5)(x + 8)^3

x = -8 and x = 5

Multiplicity of the zeros of f(x) = 9(x - 5)(x + 8)^3

x = -8 has multiplicity 3; x = 5 has multiplicity 1

Graph behavior at the zeros of f(x) = 9(x - 5)(x + 8)^3

Crosses at both (odd multiplicities)

Simplified form of f(x) = x^3 + 2x - 16x - 32

f(x) = x^3 - 14x - 32

Zeros of f(x) = x^3 - 14x - 32

x = -2, x = 1 + sqrt(11), x = 1 - sqrt(11)

x-intercepts of f(x) = x^4 - 4x^2

x = -2, 0, 2

x-values where f(x) = x^4 - 4x^2 crosses the x-axis

x = -2 and x = 2

x-values where f(x) = x^4 - 4x^2 touches and turns

x = 0

y-intercept of f(x) = x^4 - 4x^2

f(0) = 0

Type of symmetry of f(x) = x^4 - 4x^2

Even symmetry (y-axis)

Shape of the graph of f(x) = x^4 - 4x^2

W-shape

(6x^3 - 4x^2 + 3x - 6) ÷ (x - 2)

6x^2 + 8x + 19 + 32/(x - 2)

Evaluate f(x) = 6x^3 - 6x^2 - 5x + 7 at x = -3

f(-3) = -194

Zeros of f(x) = x^3 - 13x^2 + 47x - 35

1, 5, 7

All possible rational roots of f(x) = x^3 - 2x^2 - 9x + 18

±1, ±2, ±3, ±6, ±9, ±18

One rational root of f(x) = x^3 - 2x^2 - 9x + 18

x = 2

Zeros of f(x) = x^3 - 2x^2 - 9x + 18

-3, 2, 3

All possible rational roots of f(x) = x^3 - 3x^2 - 25x + 75

±1, ±3, ±5, ±15, ±25, ±75

One rational root of f(x) = x^3 - 3x^2 - 25x + 75

x = 3

Zeros of f(x) = x^3 - 3x^2 - 25x + 75

-5, 3, 5

f(2) for f(x) = x^3 - 3x^2 + 25x - 75

f(2) = -29

Vertical asymptotes and holes of f(x) = (x - 3)/(x^2 - 9)

VA: x = -3; Hole: x = 3

Horizontal asymptote of g(x) = 15x^2 / (5x^2 + 6)

y = 3

Horizontal asymptote of f(x) = 6x / (x - 2)

y = 6

Horizontal asymptote of f(x) = (x^2 + 2x - 15)/(x - 5)

No horizontal asymptote

Solve x^2 - 11x + 24 > 0

(-∞, 3) ∪ (8, ∞)

Solve x^2 ≤ 4x - 1

[2 - sqrt(3), 2 + sqrt(3)]

Simplify x ≤ 15 - 2^2

x ≤ 11

Solve (x + 7)(x - 9)/(x + 5) < 0

(-7, -5) ∪ (-5, 9)

Solve x^3 - 3x^2 - x + 3 > 0

(-∞, -1) ∪ (1, 3)