Ch. 3 - Polynomial and Rational Functions
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- Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?
Problem 1
- Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x^6-x^4+2x^2-2, we can also conclude that ƒ(1)=0
Problem 1
- Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)^4(x-3), the number 2 is a zero of multiplicity 4.
Problem 3
- Fill in the blank(s) to correctly complete each sentence. The highest point on the graph of a parabola that opens down is the ____ of the parabola.
Problem 3
- Graph each quadratic function. Give the vertex, axis, x-intercepts, y-intercept, domain, range, and largest open intervals of the domain over which each function is increasing or decreasing. ƒ(x)=-3x^2-12x-1
Problem 3
- Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.
Problem 5
- Fill in the blank(s) to correctly complete each sentence. The vertex of the graph of ƒ(x) = x^2 + 2x + 4 has x-coordinate ____ .
Problem 5
- Determine whether each statement is true or false. If false, explain why. The polynomial function ƒ(x)=2x^5+3x^4-8x^3-5x+6 has three variations in sign.
Problem 6
- Use synthetic division to perform each division. (x^3 + 3x^2 +11x + 9) / x+1
Problem 7
- Provide a short answer to each question. Is ƒ(x)=1/x^2 an even or an odd function? What symmetry does its graph exhibit?
Problem 7
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+150 between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. January
Problem 7
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+150 between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. May
Problem 7
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+150 between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. August
Problem 7
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+150 between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. October
Problem 7
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+150 between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. Sketch a graph of y=V(x) for January through December. In what month are the fewest volunteers available?
Problem 7
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+150 between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. December
Problem 7
- Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.
Problem 8
- Use synthetic division to perform each division. (5x^4 +5x^3 + 2x^2 - x-3) / x+1
Problem 9
- Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=2x^4
Problem 9
- Use the graphs of the rational functions in choices A–D to answer each question. There may be more than one correct choice. Which choices have domain (-∞, 3)U(3, ∞)?
Problem 9
- Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3-5x^2+3x+1; x-1
Problem 9
- Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3+6x^2-2x-7; x+1
Problem 10
- Use synthetic division to perform each division. (x^4 + 4x^3 + 2x^2 + 9x+4) / x+4
Problem 11
- Consider the graph of each quadratic function.(a) Give the domain and range.
Problem 11
- Use synthetic division to perform each division. (3x^3+6x^2-8x+3)/(x+3)
Problem 12
- Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x)=(x-k)q(x)+r. ƒ(x)=5x^3-3x^2+2x-6; k=2
Problem 13
- Use synthetic division to perform each division. (x^5 + 3x^4 + 2x^3 + 2x^2 + 3x+1) / x+2
Problem 13
- Consider the graph of each quadratic function.(a) Give the domain and range.
Problem 13
- Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x)=(x-k)q(x)+r. ƒ(x)=-3x^3+5x-6; k=-1
Problem 14
- Use synthetic division to perform each division. (-9x^3 + 8x^2 - 7x+2) / x-2
Problem 15
