check whether 7+3x is a factor of 3x cube +7x
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By division algorithm:-
p(x)=g(x)q(x)+r(x)
where, g(x) is the dividend,q(x) is the quotient, and r(x) is the remainder. Let r(x) here be denoted by k(any constant).
Here,
p(x)=3x³+7x=(7+3x)q(x)+k
=>p(-7/3)=-(343/9) - (49/3)={7+(-7)}q(x)+k
=>p(-7/3)= -490/9 = 0*q(x)+k
=>p(-7/3)= -490/9 =k
An expression can be the factor of a polynomial only if, on dividing that polynomial by the expression, the remainder (here, k) is equal to 0.
As k= -490/9 and k≠0
Hence, (7+3x) is not a factor of 3x³+7
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