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Factor Out - Algebra Complex Zeros

1) A polynomial P is given.
P(x) = x^4 + 49x^2
(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. If a root has multiplicity greater than one, only enter the root once.)
x =

(b) Factor P completely.
P(x) =

2) A polynomial P is given.
P(x) = x^3 â?' 10x^2 + 50x
(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. If a root has multiplicity greater than one, only enter the root once.)
x =

(b) Factor P completely.
P(x) =

3) A polynomial P is given.
P(x) = x^3 â?' 3x^2 + 4x â?' 12
(a) Factor P into linear and irreducible quadratic factors with real coefficients.
P(x) =

(b) Factor P completely into linear factors with complex coefficients.
P(x) =

4) A polynomial P is given.
P(x) = x^4 + 12x^2 â?' 64
(a) Factor P into linear and irreducible quadratic factors with real coefficients.
P(x) =

(b) Factor P completely into linear factors with complex coefficients.
P(x) =

5) Find a polynomial with integer coefficients that satisfies the given conditions.
P has degree 2 and zeros
4 + i and 4 â?' i.
P(x) =

6) Find a polynomial with integer coefficients that satisfies the given conditions.
Q has degree 3 and zeros 2,
2i, and â?'2i.
Q(x) =

Solution Preview

1) Factor out x^2:
P(x)=x^2(x^2+49)
The zeros of P(x) are x=0 with multiplicity 2 (bacuse x^2 can be written as (x-0)^2) and the zeros of x^2+49, which are x=7i, x=-7i.
So, P(x)=x^2(x-7i)(x+7i)

2)P(x) = x^3 â?' 10x^2 + 50x
Factor out x: P(x)=x(x^2-10x+50) and solve the equation x^2-10x+50=0
we get ...

Solution Summary

Factoring out is demonstrated. Complex zeros are factored out..

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