##### Ecpress g(x) as a product of linear factors from a polynomial

 Algebra Tutor: None Selected Time limit: 1 Day

For the polynomial below, -2 is a zero.

g(x)= x^3 - 2x^2 - 9x - 2

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

Oct 25th, 2015

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x = -2 is a zero of g(x).

So x + 2 is a factor of g(x).

Divide g(x) by (x + 2 ) to find the other factors

So

$g(x)=x^3-2x^2-9x-2=(x+2)(x^2-4x-1)$

Now factor $x^2-4x-1$ using the quadratic formula so that g(x) can be expressed as a product of linear factors.

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Oct 25th, 2015

a = 1, b = -4 , c = -1

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Oct 25th, 2015

Substitute values of a,b and c in the formula.

$\\ x=\frac{-(-4)\pm\sqrt{(-4)^2-4\times1\times(-1)}}{2\times1}\\ \\ x=\frac{4\pm\sqrt{16+4}}{2}\\ \\ x=\frac{4\pm\sqrt{20}}{2}\\ \\ x=\frac{4\pm2\sqrt{5}}{2}\\ \\ x=2\pm\sqrt{5}$

Oct 25th, 2015

So $x=2+\sqrt{5}$ and $x=2-\sqrt{5}$ are the zeros of the equation $x^2-4x-1$ .

Oct 25th, 2015

Oct 25th, 2015

$x^2-4x-1=(x-(2+\sqrt{5}))(x-(2-\sqrt{5}))$

So

Oct 25th, 2015

$g(x)=(x+2)(x^2-4x-1)=(x+2)(x-(2+\sqrt{5}))(x-(2-\sqrt{5}))$

expresses g(x) as a product of linear factors.

Oct 25th, 2015

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