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The polynomial x^3-3x^2-9x+c can be written in the form (x- )^

Question

The polynomial $x^{3} - 3 x^{2} - 9 x + c$ can be written in the form $(x - α)^{2} (x - β)$ , if value of $c$ is:

5

- 7

25

27

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Camila Discussed

The polynomial

x^{3} - 3 x^{2} - 9 x + c

can be written in the form

(x - α)^{2} (x - β)

, if value of

c

is:

10 mins ago

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10 mins ago

Text solutionVerified

The polynomial

x^{3} - 3 x^{2} - 9 x + c

can be written in the form of

(x - α)^{2} (x - β)

if the equation

x^{3} - 3 x^{2} - 9 x + c

has two equal roots.
Here, the roots are

α, α, β

We have

α + α + β = 3

2 α + β = 3

........... (1)
Also,

αα + α β + α β = - 9

2 α β + α^{2} = - 9

.............. (2)
Using (1) & (2), we get

2 α (3 - 2 α) + α^{2} = - 9

\Rightarrow 6 α - 3 α^{2} = - 9

\Rightarrow α^{2} - 2 α - 3 = 0

\Rightarrow (α - 3) (α + 1) = 0

\Rightarrow α = 3, - 1

when

α = - 1, β = 5

and when

α = 3, β = - 3

Also, product of roots

α^{2} β = - c

when

α = - 1, β = 5, c = - 5

and when

α = 3, β = - 3, c = 27

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Question Text	The polynomial $x^{3} - 3 x^{2} - 9 x + c$ can be written in the form $(x - α)^{2} (x - β)$ , if value of $c$ is:
Answer Type	Text solution:1
Upvotes	153