Question

State true or false: Check whether $7+3x$ is a factor of $3x^3+7x$.

• A. True
• B. False

Answer 1

The correct option is B (False)

From polynomial remainder theorem, It states that the remainder of the division of a polynomial $f\left(x\right)$ by a linear polynomial $(x-a)$ is $f\left(a\right)$.
Given linear polynomial is $7+3x=3(x+\frac{7}{3})$
$=3(x-(-\frac{7}{3})$ compare with $(x-a)$, we get $a=-\frac{7}{3}$
[or equate $7+3x$ to $0$, we would get $x=-\frac{7}{3}$]
Let $f\left(x\right)=3x^3+7x$

Consider $f\left(\frac{-7}{3}\right)=3(\frac{-7}{3}{)}^3+7(\frac{-7}{3})$

$=\frac{-343}{9}-\frac{49}{3}=\frac{-343-147}{9}=\frac{-490}{9}$

Since, $f\left(\frac{-7}{3}\right)\ne 0$

Therefore, by Factor theorem, $7+3x$ is not a factor of $3x^3+7x$