To find the possible rational roots of a polynomial, consider the factors of the constant term divided by the factors of the leading coefficient. Use synthetic division to divide the polynomial by the chosen rational root and find the quotient. Factor the quotient completely and find the x-intercepts and y-intercept. Determine the left-end and right-end behavior of the function and graph it.
To find the possible rational roots of a polynomial, we need to consider the factors of the constant term (the last term) divided by the factors of the leading coefficient (the coefficient of the highest power of x). For f(x) = 2x^3 + 3x^2 - 8x + 3, the possible rational roots are ±1, ±3, and ±1/2.
Let's take one rational root, let's say x=a. We can substitute the value of a into the polynomial and check if the result is zero. For example, if we choose a=1, we substitute x=1 into f(x) = 2x^3 + 3x^2 - 8x + 3, which gives us 2(1)^3 + 3(1)^2 - 8(1) + 3 = 0. So, x=1 is a rational root.
We can use synthetic division to divide the polynomial by (x-a). For f(x) = 2x^3 + 3x^2 - 8x + 3 and x=1, the division would be:
1 │ 2 3 -8 3
│ 2 5 -3
│_______________________
2 5 -3 0
So, the quotient q(x) is 2x^2 + 5x - 3.
To factor q(x) completely, we can use factoring techniques such as grouping or the quadratic formula. However, in this case, q(x) = 2x^2 + 5x - 3 can be factored as (2x - 1)(x + 3).
The x-intercepts of f(x) are the values of x when f(x) = 0. So, we set f(x) = 0 and solve for x. For f(x) = 2x^3 + 3x^2 - 8x + 3, the x-intercepts are x = -1/2, 1, and -3. The y-intercept is the value of f(x) when x = 0. So, we substitute x = 0 into f(x) = 2x^3 + 3x^2 - 8x + 3, which gives us f(0) = 3. Therefore, the y-intercept is (0, 3).
The left-end behavior of f(x) can be determined by looking at the leading term, which is the term with the highest power of x. In this case, the leading term is 2x^3. As x approaches negative infinity, the sign of the leading term is positive, which means the graph of f(x) goes up on the left side. The right-end behavior of f(x) can be determined by looking at the degree of the polynomial and the leading coefficient. Since the degree is odd (3) and the leading coefficient is positive (2), as x approaches positive infinity, the graph of f(x) also goes up on the right side.
We can graph y = f(x) by plotting the x-intercepts (-1/2, 0), (1, 0), and (-3, 0), the y-intercept (0, 3), and any additional points we find along the way. The graph will have a shape similar to a cubic function, going up on the left side and right side.