Use Descartes' Rule of Signs to explain why $2x^4+6x^2+8=0$ has no real roots.

For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. $f\left(x\right)=x^3+3x^2-4$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x³−8x²+x+2; between 2 and 3

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x⁴+10x³+5x²+x+1; f(−2/3)

Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x³/(3x²+1)

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 2x³−x²−9x−4=0

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=−4x²+8x−3

Use synthetic division to divide f(x)=x³−4x²+x+6 by x+1. Use the result to find all zeros of f.

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. $x^3\ge 9x^2$

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x³≤4x²

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x⁴−2x³+x²+12x+8

Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=5x²−5x

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x−4)/(x+3) > 0

Solve the equation 2x³−3x²−11x+6=0 given that -2 is a zero of f(x)=2x³−3x²−11x+6.

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x−1)

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0

Give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x⁴−3x³−20x²−24x−8=0

Solve the equation 12x³+16x²−5x−3=0 given that -3/2 is a root.

Describe in words the variation shown by the given equation. $z = \frac{k\sqrt{x}}{y^2}$

In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (−x+2)/(x−4)≥0

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=3x⁴−11x³−x²+19x+6

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=(1/x) + 2

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

Give the domain and the range of each quadratic function whose graph is described. Maximum = -6 at x = 10

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (−x−3)/(x+2)≤0

Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x² but with the given point as the vertex. (5, 3)

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (4−2x)/(3x+4)≤0