What is the remainder when the polynomial P(x) = 2x^3 - 5x^2 + 3x - 7 is divided by (x - 2)?
Polynomial Rational Functions
Question · Easy
Hundreds of balls make a bell curve
Short
What are the vertical asymptotes of the rational function f(x) = (x + 3) / (x^2 - 16)?
Solve the radical equation sqrt(3x + 1) = 4 for x.
What is the horizontal asymptote of the rational function g(x) = (6x^2 - 5) / (2x^2 + 7x + 1)?
Simplify the rational expression (x^2 - 9) / (x^2 + 5x + 6) for all valid domain values.
Given f(x) = x^2 + 2 and g(x) = 3x - 1, find the value of (f o g)(2).
What is the inverse function f^(-1)(x) for f(x) = 2x + 8?
Which cubic polynomial function has zeros at x = -1, x = 2, and x = 4?
Solve the rational equation 2 / (x - 3) + 1 / x = 6 / (x(x - 3)).
Question · Medium
Solve the radical equation sqrt(2x + 3) - x = 0.
According to the Rational Root Theorem, which list contains all possible rational roots of P(x) = 3x^3 - 2x^2 + 5x - 4?
Simplify the sum 3 / (x + 2) + 5 / (x - 4).
Divide (2x^3 - 3x^2 + 4x + 5) by (x + 1). What is the quotient and remainder?
If f(x) = sqrt(x - 3) and g(x) = x^2 + 3 for x >= 0, which statement correctly describes their relationship?
What is the domain of the function h(x) = sqrt(x + 4) / (x - 2)?
Describe the end behavior of P(x) = -3x^4 + 5x^3 - 2x + 1.
Find all real solutions to x^4 - 13x^2 + 36 = 0.
Solve the rational inequality (x - 1) / (x + 3) <= 0.
Find the slant (oblique) asymptote of f(x) = (2x^2 + 3x - 1) / (x + 1).
Determine the multiplicity of the root x = 2 for P(x) = (x - 2)^3 (x + 1)^2 (x - 5) and describe the graph's behavior at x = 2.
Find all complex zeros of P(x) = x^3 - 3x^2 + 4x - 12.
Question · Hard
Solve the radical equation sqrt(x + 7) - sqrt(x) = 1 for x.
Decompose (7x - 1) / ((x - 1)(x + 2)) into partial fractions A / (x - 1) + B / (x + 2). What are the values of A and B?
Find the inverse function f^(-1)(x) for f(x) = (3x + 1) / (x - 2).
What is the product of the complex roots of the polynomial P(x) = x^2 - 4x + 13?
Convert an angle measurement of 150 degrees to radians.
Trigonometry And Modeling
What is the exact value of sin(4pi / 3)?
What is the amplitude of the function y = -4 cos(3x - pi) + 2?
What is the period of the function f(x) = sin(4x)?
Which expression is equivalent to sin^2(theta) + cos^2(theta) for any angle theta?
What is the exact value of tan(225 degrees)?
In a right triangle, if the side opposite angle theta is 6 and the hypotenuse is 10, what is cos(theta)?
Convert 7pi / 4 radians to degrees.
Determine the phase shift of y = 3 sin(2x - pi/2).
Solve for x in the interval [0, 2pi): 2 sin(x) + 1 = 0.
Simplify (1 - cos^2(theta)) / sin(theta) for all valid theta.
A Ferris wheel has a radius of 25 feet and rotates once every 40 seconds. Its center is 30 feet above ground level. Which function models a rider's height h(t) in feet above ground as a function of time t in seconds, starting at the lowest position at t = 0?
In triangle ABC, side a = 8, side b = 10, and included angle C = 60 degrees. Find the exact length of side c.
In triangle XYZ, angle X = 45 degrees, angle Y = 60 degrees, and side x = 12. Find side y.
Evaluate cos(75 degrees) using the sum identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B) with A = 45 degrees and B = 30 degrees.
What is the range of f(x) = -3 sin(2x) + 5?
Simplify sec(theta) * cot(theta) * sin(theta).
Solve cos(2x) = 0 for x in [0, pi).
If sin(theta) = 3/5 and theta is in Quadrant II, what is cos(theta)?
Evaluate sin(2 theta) given cos(theta) = -5/13 and theta is in Quadrant III.
Solve 2 cos^2(x) - cos(x) - 1 = 0 for x in [0, 2pi).
Tide height in feet is modeled by h(t) = 4 sin(pi/6 (t - 2)) + 7 where t is hours after midnight. What is the maximum water height and at what time t between 0 and 12 hours does it first occur?
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