Question 1

How many and of which kind of roots does the equation f(x) = x^4 - 2x³ - 11x² + 12x + 36 have?OA. 4 realB. 2 real; 2 complexOC. 4 real; 2 complexD. 3 realReset Selection

How many and of which kind of roots does the equation f(x) = x^4 - 2x³ - 11x² + 12x-example-1

Question 2

ANSWER:

Explanation:

We have the following function:

$f(x)=x^4\:-\:2x³\:-\:11x²\:+\:12x\:+\:36$

We solve the polynomial as follows:

$\begin{gathered} x^4-2x^3-11x^2+12x+36=0 \\ \\ \left(x+2\right)(x^4-2x^3-11x^2+12x+36)/(x+2) \\ \\ \begin{matrix}\texttt{\:\:\:-2¦\:\:\:\:1\:\:\:-2\:\:-11\:\:\:12\:\:\:36}\\ \texttt{\:\:\:\:\:¦\underline{\:\:\:\:\:\:\:\:-2\:\:\:\:8\:\:\:\:6\:\:-36}}\\ \texttt{\:\:\:\:\:\:\:\:\:\:1\:\:\:-4\:\:\:-3\:\:\:18\:\:\:\:0}\end{matrix} \\ \\ (x^4-2x^3-11x^2+12x+36)/(x+2)=x^3-4x^2-3x+18 \\ \\ \left(x+2\right)(x^3-4x^2-3x+18)/(x+2) \\ \\ \begin{matrix}\texttt{\:\:-2¦\:\:\:1\:\:-4\:\:-3\:\:18}\\ \texttt{\:\:\:\:¦\underline{\:\:\:\:\:\:-2\:\:12\:-18}}\\ \texttt{\:\:\:\:\:\:\:\:1\:\:-6\:\:\:9\:\:\:0}\end{matrix} \\ \\ (x^3-4x^2-3x+18)/(x+2)=x^2-6x+9 \\ \\ (x+2)(x+2)(x^2-6x+9) \\ \\ (x^2-6x+9)=(x-3)^2 \\ \\ (x+2)^2(x-3)^2=0 \\ \\ x+2=0\rightarrow x=-2 \\ \\ x-3=0\operatorname{\rightarrow}x=3 \end{gathered}$

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