The values for x when f(x) = 0 are:
x=-2 and x=3
My question is:
The function f(x) can be written in the form of a(x-p)(x-q)(x-r)
Write down one possible set of values for p, q, and r?
Now Ive got the correct answer x = -3, 2 and 2
Could you show me step by step how I got that answer?
The values for x when f(x) = 0 are:
x=-2 and x=3
My question is:
The function f(x) can be written in the form of a(x-p)(x-q)(x-r)
Write down one possible set of values for p, q, and r?
Now Ive got the correct answer x = -3, 2 and 2
Could you show me step-by step how to end up with those values?
Do you substitute p, q and r with 0 .
When f(-2)=0 means that (x+2) is a factor of the polynomial.
Similarly, when f(3)=0 means that (x-3) is a factor of the polynomial.
Let a be a mulplicative constant (undetermined), then f(x) is of the form:
f(x) = a(x+2)(x-3)....
Since the question tells us that the function is a cubic, i.e. has three factors, and there are only two distinct roots, the third root must be identical to one of the other two, i.e. the third factor must be either (x+2) or (x-3).
So the two possible answers are:
f(x) = a(x+2)^2(x-3)
or
....
Can you figure out the other one?
By the way, you can look at this link for a sketch of the functions:
http://img41.imageshack.us/img41/9449/adrianat.png
Sure! To find the possible values for p, q, and r in the equation form a(x-p)(x-q)(x-r), we can use the given values where f(x) = 0, which are x=-2 and x=3.
Step 1: Plug in the values x=-2 and x=3 into the equation:
0 = a(-2 - p)(-2 - q)(-2 - r)
0 = a(3 - p)(3 - q)(3 - r)
Step 2: Simplify the equation:
Now we have two equations:
0 = a(-2 - p)(-2 - q)(-2 - r)
0 = a(3 - p)(3 - q)(3 - r)
Step 3: Expand the equation:
0 = a(-2p - 2q - 2r + 4pq + 4pr + 4qr - 8pqr)
0 = a(9 - 3p - 3q + pq - 3r + pr + qr - prq)
Step 4: Equate the expanded equations:
-2p - 2q - 2r + 4pq + 4pr + 4qr - 8pqr = 9 - 3p - 3q + pq - 3r + pr + qr - prq
Step 5: Collect like terms:
-8pqr + 4pq - pq + pr + 4qr - qr - 3p - 3q + 3r = 9 - 2p - 2q
Step 6: Move all terms to one side:
-8pqr + pq + pr + qr - p - q + r = 9 - 2p - 2q - 3r
Step 7: Compare the coefficients of each term on both sides of the equation. From the coefficients of p, q, and r, we can determine the values for p, q, and r:
Coefficients of p: -8pqr = -2p
-8qr = -2
qr/4 = 1/4
Coefficients of q: -8pqr = -2q
-8pr = -2
pr/4 = 1/4
Coefficients of r: -8pqr = -3r
-8pq = -3
pq/8 = 3/8
Step 8: Solve the simultaneous equations:
qr/4 = 1/4 => qr = 1
pr/4 = 1/4 => pr = 1
pq/8 = 3/8 => pq = 3
Step 9: Find the values of p, q, and r:
From the above equations, we can choose values such that:
qr = 1 => q=1, r=1
pr = 1 => p=1, r=1
pq = 3 => p=3, q=2
Therefore, one possible set of values for p, q, and r is p=3, q=2, and r=1.