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Answer:
P(x) = ()x(x + 3)(x - 1)²
Step-by-step explanation:
Given - The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=1 and roots of multiplicity 1 at x=0 and x=-3. It goes through the point (5,256)
To find - Formula for P(x) ?
Proof -
Given that,
P(x) has a root of multiplicity 2 at x=1
So,
(x - 1)² is a factor.
Now,
Given that, there is roots of multiplicity 1 at x=0 and x=-3
So,
(x - 0) and ( x - (-3) are also factor
So,
P(x) can be written as
P(x) = Ax(x + 3)(x - 1)² .........(1)
where A is a constant.
Now,
Given that, It goes through the point (5,256)
⇒At x = 5, P(x) = 256
So,
Put the values of x and P(x) in equation (1), we get
P(x) = Ax(x + 3)(x - 1)²
⇒256 = A(5)(5 + 3)(5 - 1)²
⇒256 = 5A(8)(4)²
⇒256 = 40A(16)
⇒256 = 640 A
⇒A = =
∴ we get
P(x) = ()x(x + 3)(x - 1)²