# 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..