The function f is defined by f(x) = 2x³ + 3x² + cx + 8, where c is a constant. In the xy-plane, the graph of f intersects the x-axis at the three points (−4, 0), (1/2, 0), and (p, 0). What is the value of c?
-18
-2
2
10
ok, the bot is wrong,
using (−4, 0)
f(-4) = 2(-4)³ + 3(-4)² + c(-4) + 8 = 0
= -128 + 48 - 4c + 8
= 0
then c = -18
and f(x) = 2x³ + 3x² - 18x + 8
check for f(1/2) = 0
2(1/8) + 3(1/4) - 18(1/2) + 8 = 0
c = -18
don't know what p has to do with the question.
thank you smm
To find the value of c, we need to substitute the given points of intersection with the x-axis into the equation and solve for c.
Given points of intersection with the x-axis are:
(−4, 0)
(1/2, 0)
(p, 0)
For the point (-4, 0), we can substitute x = -4 into the equation f(x) = 2x³ + 3x² + cx + 8 and solve for c:
f(-4) = 2(-4)³ + 3(-4)² + c(-4) + 8
0 = -128 - 48 - 4c + 8
0 = -128 - 40 - 4c
0 = -168 - 4c
For the point (1/2, 0), substituting x = 1/2 into the equation:
f(1/2) = 2(1/2)³ + 3(1/2)² + c(1/2) + 8
0 = 1/2 + 3/4 + c/2 + 8
0 = 4/4 + 3/4 + c/2 + 8
0 = 7/4 + c/2 + 8
For the point (p, 0), where p is the x-coordinate, we substitute it into the equation:
f(p) = 2p³ + 3p² + cp + 8
0 = 2p³ + 3p² + cp + 8
We have three equations derived from the given points of intersection:
-168 - 4c = 0 ... Equation 1
7/4 + c/2 + 8 = 0 ... Equation 2
2p³ + 3p² + cp + 8 = 0 ... Equation 3
To solve this system of equations, we can use substitution or elimination method. We'll use the elimination method.
From Equation 1, we isolate c:
-4c = 168
c = -42
Substituting c = -42 into Equation 2:
7/4 - 42/2 + 8 = 0
7/4 - 21 + 8 = 0
7/4 - 84/4 + 8 = 0
-77/4 + 32/4 = 0
-45/4 = 0
Since -45/4 is not equal to 0, the equation is inconsistent. This means there is no value of p that satisfies the third equation.
Therefore, the value of c cannot be determined from the given information.