Respuesta :

danielmaduroh
We have the following polynomial:

[tex]f(x)=2x^{3}+9x^{2}+7x-6[/tex]

The problem states that one root is -3. Thus, it is true that [tex](x+3)[/tex] is a factor of the polynomial. Given that this is fulfilled, it is also true that:

[tex]f(x)=(x+3)Q(x) \therefore Q(x)=\frac{f(x)}{x+3} \\ \\ where \ Q(x) \ has \ a \ degree \ of \ 2[/tex]

We can find Q(x) by applying Ruffini's rule, thus:

[tex]\ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ 9 \ \ \ \ \ \ \ \ 7 \ \ \ \ \ \ -6 \\ -3 \\ \rule{50mm}{0.1mm} \\ \ {} \ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ 3 \ \ \ \ \ -2 \ \ \ \ \ \ \ \ 0[/tex]

Therefore:

[tex]Q(x)=2x^{2}+3x-2[/tex]

The roots of this polynomial can be get as follows:

[tex]x_{12}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \rightarrow x_{12}=\frac{-3\pm\sqrt{3^2-4(2)(-2)}}{2(2)}\\x_{1}=\frac{1}{2};\ x_{2}=-2[/tex]

These are the roots along with [tex]-3[/tex]. Finally, the factored polynomial can be written as follows:

[tex]f(x)=(x+3)(x+2)(2x-1)[/tex]
kenchellbegay

Identify the factor x + 3 from the given root –3.

Use synthetic division to divide the polynomial by x + 3.

Use the bottom row of the synthetic division as coefficients in the quadratic 2x2 + 3x – 2.

Factor the quotient to find the two other factors: x + 2 and 2x – 1.

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