Question

For each of the following functions f(x)=2x3+3x2-8x+3, f(x)=x3-3x+2, f(x)=x4-x3-2x-4 answer the following questions.

1. List all possible rational roots of f(x)
2. Find one rational root x=a
3. Use long division (synthetic division) to find the quotient q(x), where f(x)=(x-a)q(x).
4. Use rational root test or factoring to factor completely q(x).
5. List all x-intercepts of f(x) and y-intercept.
6. State left-end and right-end behavior of f(x)
7. Graph y=f(x), show all additional points that were used for graphing.

Answer 1

To find the rational roots of a polynomial function, we can use the rational root theorem. Using long division or synthetic division, we can find the quotient and factor it completely. The x-intercepts and y-intercept can be found by solving the equation f(x)=0 and substituting x=0, respectively. The left-end and right-end behavior of the function can be determined by looking at the leading term and coefficient. Graphing the function involves plotting the x-intercepts, y-intercept, and using the end behaviors.

Step-by-step explanation:

1. To find the possible rational roots of the function f(x), we use the rational root theorem. The possible rational roots will be of the form ±p/q, where p is a factor of the constant term (in this case 3) and q is a factor of the leading coefficient (in this case 2). So the possible rational roots are ±1, ±3, ±1/2, ±3/2.
2. To find one rational root x=a, we can try substituting each of the possible rational roots into the function and check if f(a)=0. For example, if we substitute x=1 into f(x)=2x^3+3x^2-8x+3, we get f(1)=2(1)^3+3(1)^2-8(1)+3=0. So x=1 is a rational root.
3. To use long division or synthetic division to find the quotient q(x), we divide the function f(x) by (x-a), where a is the rational root we found in the previous step. For example, if we divide f(x)=2x^3+3x^2-8x+3 by (x-1), we get the quotient q(x)=2x^2+5x-3.
4. To factor the quotient q(x) completely, we can use factoring techniques such as grouping or the quadratic formula. In this case, the quotient q(x)=2x^2+5x-3 can be factored as (2x-1)(x+3).
5. To find the x-intercepts of f(x), we set f(x) equal to zero and solve for x. For example, if we set f(x)=2x^3+3x^2-8x+3 equal to zero, we can solve for x by factoring or using numerical methods such as the quadratic formula or graphing calculator. The y-intercept is the value of f(x) when x=0. So we substitute x=0 into f(x) and find the y-intercept.
6. The left-end behavior of f(x) can be determined by looking at the leading term of the function. In this case, the leading term is 2x^3. As x approaches negative infinity, the leading term becomes negative and the function goes to negative infinity. The right-end behavior can be determined by looking at the degree of the polynomial and the sign of the leading coefficient. In this case, the leading coefficient is positive and the degree is odd, so as x approaches positive infinity, the function goes to positive infinity.
7. To graph y=f(x), we can plot points using the x-intercepts, y-intercept, and any additional points we found in previous steps. We can also use the end behaviors to determine the shape of the graph.