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

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For the polynomial below -3 is a zero

f(x)= x^3 + 9x^2 + 24x + 18

Express f(x) as a product of linear factors

Oct 18th, 2015

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x = -3 is a zero of the polynomial

So x + 3 is a factor.

Divide f(x) by (x+3) to find the other factors.

So

$f(x)=x^3+9x^2+24x+18=(x+3)(x^2+6x+6)$

Now factorize $x^2+6x+6$  since we need linear factors. We can use the quadratic formula to do that.

a = 1, b = 6, c = 6

$\\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ \\ x=\frac{-6\pm \sqrt{6^2-4\times1\times6}}{2\times1}\\ \\ x=\frac{-6\pm \sqrt{36-24}}{2}\\ \\ x=\frac{-6\pm \sqrt{12}}{2}\\ \\ x=\frac{-6\pm 2\sqrt{3}}{2}\\ \\ x=-3\pm \sqrt{3}\\$

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Oct 18th, 2015

sdf

Oct 18th, 2015

Remaining part of the answer is given below.Sorry for the delay.

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So

$(x^2+6x+6)=(x-(-3+\sqrt{3})(x-(-3-\sqrt{3})))=(x+(3-\sqrt{3}))(x+(3+\sqrt{3}))$

Oct 18th, 2015

Hence f(x) as a product of linear factors is$f(x)=(x+3)(x+(3-\sqrt{3}))(x+(3+\sqrt{3}))$

Oct 18th, 2015

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Oct 18th, 2015
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Oct 18th, 2015
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