2.5: Encontrar factores desde las raíces
- Page ID
- 111886
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Un método para resolver ecuaciones implica encontrar los factores de la expresión polinómica en la ecuación y luego establecer cada factor igual a cero.
\ [
\ begin {array} {c}
x^ {2} +8 x+15=0\\
(x+5) (x+3) =0\\
x+5=0\ quad x+3=0\\
x=-5\ quad x=-3
\ end {array}
\]
En este proceso, el razonamiento es que si\((x+5)\) veces \((x+3)\)es igual a cero, entonces una de esas expresiones debe ser igual a cero. Al establecerlos iguales a cero, encontramos las soluciones de\(x=-5,-3 .\) Plugging them back into the factorizado expression we see the following:
\ [
(-5+5) (-5+3) =0 *-2=0
\]
y
\ [
(-3+5) (-3+3) =2 * 0=0
\]
Este proceso también funciona a la inversa. Es decir, si conocemos una raíz de la función, podemos encontrar factores para la expresión.
Ejemplo
Encuentra una ecuación cuadrática que tenga raíces de -2 y +3
\ [
\ begin {array} {cc}
x=-2 & x=3\\
x+2=0 & x-3=0\\
(x+2) (x-3) =0\\
x^ {2} -x-6=0
\ end {array}
\]
Las raíces que son fracciones son un poco más complicadas, pero realmente no más difíciles:
Ejemplo
Encuentra una ecuación cuadrática que tenga raíces de -5 y\(\frac{2}{3}\)
\ [\ begin {array} {cc}
x=-5 & x=\ frac {2} {3}\\
x+5=0 & 3 x=2\\
x+5=0 & 3 x-2=0\\
(x+5) (3 x-2) =0\\
3 x^ {2} +13 x-10=0
\ end {array}
\]
Ejercicios 2.5
Encuentra una ecuación cuadrática que tenga las raíces indicadas.
1)\(\quad 4,-1\)
2)\(\quad -2,7\)
3)\(\quad \frac{3}{2}, 1\)
4)\(\quad-\frac{1}{5}, \frac{2}{3}\)
5)\(\quad \frac{1}{3}, 3\)
6)\(\quad-4, \frac{2}{5}\)
7)\(\quad \frac{1}{2},-\frac{7}{2}\)
8)\(\quad-1, \frac{3}{5}\)
9)\(\quad-\frac{2}{3},-3\)
10)\(\quad-\frac{2}{3},-\frac{3}{4}\)
11)\(\quad-\frac{5}{2}, 3\)
12)\(\quad-6,-2\)