9514 1404 393
Answer:
a) k = -19
b) P(x) = (5x +1)(3x -1)(2x -1)
c) x = {-1/5, 1/3, 1/2}
d) see attached
e) no; P(-1) = -48
f) see attached
Step-by-step explanation:
a) The second attachment shows synthetic division of P(x) by the factor (x -1/3). The remainder is 0 when the final expression is zero:
1 + (10 +k)/9 = 0
(10 +k)/9 = -1 . . . . . subtract 1
10 +k = -9 . . . . . . . .multiply by 9
k = -19 . . . . . . . . . . subtract 10
For (3x -1) to be a factor, the value of k is -19.
__
b) The graph (see part C) shows the zeros of the function to be -0.2, 0.333..., and 0.5. This means the linear factors are ...
P(x) = (5x +1)(3x -1)(2x -1)
The linear factors can also be obtained by finishing the factoring of ...
P(x) = (3x -1)(10x^2 -3x -1)
The quadratic factor is the synthetic division result, divided by 3. It can be factored by grouping: 10x^2 -3x -1 = (10x^2 -5x) +(2x -1) = 5x(2x -1) +1(2x -1).
So P(x) = (3x -1)(5x +1)(2x -1).
__
c) The graph shows the zeros to be x ∈ {-1/5, 1/3, 1/2}.
__
d) See the attachment for a graph
__
e) The point (-1, -40) is not on the graph. Evaluating P(-1) gives -48. The point that would be on the graph for x=-1 is (-1, -48).
__
f) See the attachment.
