Question
Check whether $7 + 3x $ is a factor of $ 3{x^3} + 7x$.
✓
Answer
We know that if the polynomial $7+3x$ is a factor of $ 3{x^3} + 7x$, then on dividing the polynomial $3{x^3} + 7x$ by $7+3x$, we must get the remainder as $0.$
We need to find the zero of the polynomial $7 + 3x$
$\begin{gathered} 7 + 3x = 0 \hfill \\ \Rightarrow {\text{ }}x = - \frac{7}{3} \hfill \\ \end{gathered} $
While applying the remainder theorem, we need to put the zero of the polynomial $7+3x$ in the polynomial $ 3{x^3} + 7x$,to get
$\,\,p\left( x \right) = 3{x^3} + 7x$
$p\left( {{{ - 7} \over 3}} \right)$ $= 3{\left( { - \frac{7}{3}} \right)^3} + 7\left( { - \frac{7}{3}} \right)\,\, = 3\left( { - \frac{{343}}{{27}}} \right) - \frac{{49}}{3}$
$ = - \frac{{343}}{9} - \frac{{49}}{3}\,\, = \frac{{ - 343 - 147}}{9}$
$= \frac{{ - 490}}{9}.$
We conclude that on dividing the polynomial $3{x^3} + 7x $ by $7 + 3x$, we will get the remainder as $\frac{{ - 490}}{9}$ which is not $0.$
Therefore, we conclude that $7 + 3x$ is not a factor of $ 3{x^3} + 7x$
We need to find the zero of the polynomial $7 + 3x$
$\begin{gathered} 7 + 3x = 0 \hfill \\ \Rightarrow {\text{ }}x = - \frac{7}{3} \hfill \\ \end{gathered} $
While applying the remainder theorem, we need to put the zero of the polynomial $7+3x$ in the polynomial $ 3{x^3} + 7x$,to get
$\,\,p\left( x \right) = 3{x^3} + 7x$
$p\left( {{{ - 7} \over 3}} \right)$ $= 3{\left( { - \frac{7}{3}} \right)^3} + 7\left( { - \frac{7}{3}} \right)\,\, = 3\left( { - \frac{{343}}{{27}}} \right) - \frac{{49}}{3}$
$ = - \frac{{343}}{9} - \frac{{49}}{3}\,\, = \frac{{ - 343 - 147}}{9}$
$= \frac{{ - 490}}{9}.$
We conclude that on dividing the polynomial $3{x^3} + 7x $ by $7 + 3x$, we will get the remainder as $\frac{{ - 490}}{9}$ which is not $0.$
Therefore, we conclude that $7 + 3x$ is not a factor of $ 3{x^3} + 7x$
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