The characteristics of the graph depends on the leading coefficient and
the degree of the polynomial.
Responses:
a. From the positive leading polynomial of the odd degree polynomial;
b. x = 4, x = 0.5, and x = -3
c. Yes the graph crosses the x-axis
d. The y-intercept is the point y = 12
e. 2 turning points
f. Additional points of the graph are(1, -12), (3, -30) and (5, 72)
g. Please find attached the graph of the function.
The given function is; P(x) = 2·x³ - 3·x² - 23·x + 12
a. Given that the leading coefficient is positive and the degree is odd, we have;
b. The zeros of the function are found as follows;
P(x) = 2·x³ - 3·x² - 23·x + 12
At the zeros, 2·x³ - 3·x² - 23·x + 12 = 0
By factorization, using an online tool, we have;
2·x³ - 3·x² - 23·x + 12 = (x - 4)·(2·x² + 5·x - 3) = (x - 4)·(2·x - 1)·(x + 3) = 0
Alternatively, we have;
By trial and error, we have, at x = 4, 2·4³ - 3·4² - 23·4 + 12 = 0
Therefore;
(x - 4) is a factor
(2·x³ - 3·x² - 23·x + 12) ÷ (x - 4) = (2·x² + 5·x - 3)
(2·x² + 5·x - 3) = 2·x² + 6·x - x - 3 = 2·x·(x + 3) - 1(x + 3) = 0
(2·x² + 5·x - 3) = 2·x·(x + 3) - 1(x + 3) = (2·x - 1)·(x + 3)
Which gives;
2·x³ - 3·x² - 23·x + 12 = (x - 4)·(2·x - 1)·(x + 3)
Therefore, the zeros are;
x = 4, x = [tex]\frac{1}{2}[/tex] = 0.5, and x = -3
c. The maximum number of turning points in a cubic function are 2
Given that the graph has three zeros at x = -3, x = 0.5, and x = 4, the
graph crosses the x-axis to get three zeros from the two turning points.
d. The y-intercept is given by the point at which the graph crosses the y-axis, which is the point, x = 0 which is found as follows;
f(0) = 2·0³ - 3·0² - 23·0 + 12 = 12
Which gives the point (0, 12)
e. The number of turning points of a polynomial of degree n is n - 1
Therefore;
f. Additional points of the graph are;
f(1) = 2·1³ - 3·1² - 23·1 + 12 = -12
f(3) = 2·3³ - 3·3² - 23·3 + 12 = -30
f(5) = 2·5³ - 3·5² - 23·5 + 12 = 72
Therefore;
The points, (1, -12), (3, -30) and (5, 72) are points on the graph
g. Please find attached the graph of the function created with MS Excel
Learn more about the graphs of functions here:
https://brainly.com/question/14387154
