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23 Polynomials Problems : Solving for Roots, Asymptotes, Word Problems, Finding Equations from Roots, Synthetic Division and Function Composition

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1. The figure shows the graphs of f(x) = X 3 and g(x) =AX 3.What can you conclude about the value of a?

2. If f(x)= x(x+3)(x-1), use interval notation to give all values of x where f(x)>0.

3. If f(x) =x(x-1)(x-4)2 , use interval notation to give all values of x where f(x)>0.

4. Find the quotient and remainder of f(x) = X 3 -4 X2 + 5x +5 divided by p(x) = x-1.

5. Find the quotient and remainder of f(x) = X4 -2 divided by p(x) = x-1.

6. Find a polynomial with the leading coefficient 1 and degree 3 that has-1,1and 3 as roots .

7. The polynomial f (x) divided x-3 results in a quotient of X 2 + 3x-5 with a remainder of 2.find f(3).

8. Let f(x)= X3 - 8X2 + 17x-9. Using the factor theorem to find other solutions to F(x)-F(1)=0, besides x=1

9. Find the polynomial F(x) of degree three that has zeroes at 1, 2, and 4 such that F(0)= -16.

10. The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3=I among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.

11. Given that (3x-a)(x-2)(x-7)=3X3-32X2 + 81x-70, determine the value of a.

12. Find the roots of the polynomial X 3 - X2 + 16x-16.

13. Find the vertical asymptote of the rational function f(x)=3x-12/4x-2

14. Find the horizontal asymptote of the rational function f(x)= 8x-12/4x-2

15. For f(x)= 1/ X2-2x-8, find the interval(s)where f(x)<0.

16. Express the following statement as a formula with the value of the constant of proportionality determined with the given conditions: w varies directly as x and inversely as the square of y. If x=15 and y=5, then w=36.

17. The electrical resistance R of a wire varies directly as its length L and inversely as the square of its diameter. A wire 20 meters long and 0.6 centimeters in diameter made from a certain alloy has a resistance of 36 ohms. What is the resistance of a piece of wire 60 meters long and 1.2 centimeters in diameter made from the material?

18. The period of a simple pendulum is directly proportional to the square root of its length. If the pendulum has a length of 6 feet and a period of 2 seconds to what length should it be shorten to achieve a 1 second period?

19. Find the equation for the line with y-intercept 3 that is perpendicular to the line y = 2/3 x-4.

20. If P(4,-5) is a point on a graph of the function y = f(x), find the corresponding point on the graph of y =2f(x-6).

21. Explain how the graph of y-5=(x-3)2 can be obtained from the graph of y=X 2.

22. An object is projected upward from the top of a tower. It's the distance in feet above the ground after t seconds is given by s(t) = -16t 2 + 64t +80. How many second will it take to reach the ground level.

23. Given f(x) = 5x +7 and g(x) =X2 +7 , find (g o f)(x).

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Solving for Roots, Asymptotes, Word Problems, Finding Equations from Roots, Synthetic Division and Function Composition are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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