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german
4 years ago
9

Factor the expression: -\lambda ^{3}+4\lambda ^{2}+4\lambda -16

Mathematics
1 answer:
Anit [1.1K]4 years ago
3 0

Answer:

-(\lambda -4)(\lambda -2)(\lambda +2)

Step-by-step explanation:

What we want to fator is:

-\lambda^3 + 4 \lambda^2 + 4\lambda - 16

There is no common factor, but let's factor it by grouping. The first two addends can be factor as follows:

-\lambda^3 + 4 \lambda^2 = \lambda^2(-\lambda + 4) = -\lambda^2(\lambda - 4)

the second addends can be factor as well:

4\lambda - 16 = 4(\lambda- 4).

Then our original expression can be rewritten like

-\lambda^3 + 4 \lambda^2 + 4\lambda -16=\lambda^2(\lambda - 4) + 4(\lambda - 4)

And here the (\lambda-4) is the common factor!

-\lambda^2(\lambda - 4) + 4(\lambda - 4) = (\lambda - 4)(-\lambda^2 + 4)

Finally, we can factor the quadratic expression as a difference of squares -\lambda^2 + 4 = 4 - \lambda^2 = (2+\lambda)(2-\lambda)

Ant we get

(\lambda - 4)(-\lambda^2 + 4)= (\lambda - 4)(\lambda + 2)(2-\lambda)

now, we can extract the negative sign from (2-\lambda), and we get

-(\lambda -4)(\lambda -2)(\lambda +2).

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