Simple Polynomial Question (Special Results) (1 Viewer)

[ThreeOne]

I just need a little help with this question, it won't take long. Thanks in advance.

A polynomial P(x) of degree 3 has a triple root at x=1.
Find P(x). Is this a unique answer?

The answer is p(x)=k(x-1)^3

What I don't understand is why there a is "k" in the answer.

Oh, and what does a "unique answer" mean?

 

[Riviet]

Unique answer in this question refers to a "unique polynomial", ie the one and only polynomial that satisifies the condition(s) in the question.

The k simply makes the curve steeper or shallower or changes the concavity of the curve (if k is negative), whilst not really affecting anything else in the polynomial.

e.g y=9999(x-1)³ and y=0.0001(x-1)³ both have triple roots at x=1.

 

[ThreeOne]

"A monic polynomial P(x) of degree 2 has a double root at x=-4. Find P(x)."

The answer to that doesn't have a k. Why?

 

[Riviet]

"A monic polynomial P(x) of degree 2 has a double root at x=-4. Find P(x)."

The answer to that doesn't have a k. Why?

A monic polynomial is a polynomial which the coefficient of the leading term (term with highest power) being 1.

e.g g(x) = 1x² + 2x - 7

Solution to question which also shows why k disappears:
Let P(x)1x² + bx + c, since P(x) is monic where b and c are real.

Consider k(x+4)² = 1x² + bx + c,

Expanding, kx² + 8kx + 16 = 1x² + bx + c

Equating coeffiecents of x², x and constant term,

k=1,

8k=b
.'. b=8

c=16

.'. P(x) = 1x² + bx + c
=x² + 8x + 16
=(x+4)²

Hence the solution that satisifies the conditions is a unique polynomial => no k term required.