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If f(x) = 2x^3 + 10x^2 + 18x + 10 and x + 1 is a factor of f(x), then find all of the zeros of f(x) algebraically

Sagot :

Given the polynomial:

[tex]f(x)=2x^3+10x^2+18x+10[/tex]

We know that (x + 1) is a factor of f(x). We divide f(x) by (x + 1):

Then:

[tex]f(x)=(x+1)(2x^2+8x+10)=2(x+1)(x^2+4x+5)[/tex]

For the quadratic term, we solve the following equation:

[tex]x^2+4x+5=0[/tex]

Using the general solution for quadratic equations:

[tex]\begin{gathered} x=\frac{-4\pm\sqrt{4^2-4\cdot1\cdot5}}{2\cdot1}=\frac{-4\pm\sqrt{16-20}}{2}=\frac{-4\pm\sqrt{4}}{2} \\ \\ \therefore x=-2\pm i \end{gathered}[/tex]

The zeros of f(x) are:

[tex]\begin{gathered} x_1=-1 \\ \\ x_2=-2-i \\ \\ x_3=-2+i \end{gathered}[/tex]

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