Answer:
The equation intersects the x or y axis a total ot three times:
(-1.5,0), (5,0), and (-15,0).
Explanation:
I will assume that the "all points of intersection" refers to the intersection of the line with the x and y axes. One can find those points by either a mathematical or graphing approach.
1) Mathematical
We and to find all points that have a zero for either the x or y coordinate: (0,y) and (x,0)
(x,0): Solve p(x)=(2x+3)(x-5) for p(x) = 0:
0=(2x+3)(x-5)
(2x+3)=0 and (x-5)=0
(2x+3)=0
2x=-3
x= - 1.5 (-1.5,0)
and
(x-5)=0
X=5 (5,0)
The two points marking the intersection with the x axis are (-1.5,0) and (5,0).
--
(0,y): Solve p(x)=(2x+3)(x-5) for p(x) for x=0:
p(x) = (2x+3)(x-5)
p(x) = (2*0+3)(0-5)
p(x) = (3)*(-5)
p(x) = -15 (-15,0)
The equation crosses the y axis at one point: (-15,0).
2. Graphing
See the attached graph. The points of intersection can be located, as shown. They match the values obtained mathematically, but a free graphing utility (DESMOS) makes graphing a pleasant chore.