Using a given zero to write a polynomial as a product of linear factors.

Algebra
Tutor: None Selected Time limit: 1 Day

For the polynomial below, the -2 is a zero.

f(x)= x^3 - 2x^2 - 6x + 4

Express f(x) as a product of linear factors.

Use fractions or radicals in answers not decimals.

Oct 17th, 2015

Hello!

Check the root -2: f(-2)=-8-8+12+4=0.

Then divide f(x) by (x+2):

x^3-2x^2-6x+4 = x^2(x+2)-2x^2-2x^2-6x+4 = x^2(x+2)-4x^2-6x+4 = x^2(x+2)-4x(x+2)+8x-6x+4 =

x^2(x+2)-4x(x+2)+2x+4 = x^2(x+2)-4x(x+2)+2(x+2) = (x+2)(x^2-4x+2).

To divide (x^2-4x+2) find its zeros, this is the quadratic equation:

x1,2 = 2+-sqrt(4-2) = 2+-sqrt(2).  (I used formula for the even second coefficient).

The answer: f(x) = (x+2)(x-2-sqrt(2))(x-2+sqrt(2)).

Please ask if something is unclear.
Oct 17th, 2015

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