Math CentralQuandaries & Queries


Question from gagan, a student:

find all the roots of z^5-3z^4+2z^3+z^2-3z+2


Many times the key to solving an algebra problem is recognizing a pattern. It's like hearing the first few bars of a song for the first time and knowing immediately the group that is performing it. That skill you get from hearing the group so many time s that you easily recognize its sound. The same is true of algebra. If you have worked with algebraic expressions often enough you see patterns.

What I see in The expression z5 - 3z4 + 2z3 + z2 - 3z + 2 is

(z5 - 3z4 + 2z3) + (z2 - 3z + 2).

I have grouped the terms in this way because the coefficients match, 2, -3 and 2. Hence I can write the expression

z3 (z2 - 3z + 2) + (z2 - 3z + 2)

and see (z2 - 3z + 2) as a common factor so that the expression becomes

(z3 + 1)(z2 - 3z + 2)

Can you complete the problem from here?

Write back if you need more assistance,


Hi Gagan. Look for factoring opportunities first.

See the coefficients: 1, -3, 2, 1, -3, 2. There's a repeated group, so
we can factor:

(z2 - 3z + 2) (z3 + 1)

Now can you finish factoring and solve the problem?

Write back if you need more help.
Stephen La Rocque

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