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

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For the polynomial below, 3 is a zero.

f(x)= 3x^3 - 10x^2 + x + 6

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

Use fractions or square roots, not decimals.

Oct 17th, 2015

Hello!

Test: f(3) = 81-90+3+6=0.

f(x) = 3x^3 - 10x^2 + x + 6 = 3x^2(x-3) + 9x^2 - 10x^2 + x + 6 = 
3x^2(x-3) - x^2 + x + 6 = 3x^2(x-3) - x(x-3) - 3x + x + 6 = 
3x^2(x-3) - x(x-3) - 2x + 6 = 3x^2(x-3) - x(x-3) - 2(x-3) =
(x-3)(3x^2-x-2).

Now solve the quadratic equation (3x^2-x-2)=0,
x1,2 = (1+-sqrt(25))/6 = 1 and -2/3.

The answer: f(x) = (x-3)(x-1)(3x+2).  (  3x+2 is 3*(x+2/3)  )

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

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Oct 17th, 2015
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Oct 17th, 2015
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