Question

# f(x)=2x3+6x2+4xg(x)=x2+3x+2The polynomials f(x) and g(x) are defined above. Which of the following polynomials is divisible by 2x+3 ?

A
h(x)=f(x)+g(x)
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B
p(x)=f(x)+3g(x)
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C
r(x)=2f(x)+3g(x)
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D
s(x)=3f(x)+2g(x)
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Solution

## The correct option is B p(x)=f(x)+3g(x)Put 2x+3=0, we get x=−32.Substitute x=−32 in f(x)=2x3+6x2+4x as follows:f(x)=2x3+6x2+4xf(−32)=2(−32)3+6(−32)2+4(−32)f(−32)=2(−278)+6(94)−6 f(−32)=−274+544−6 f(−32)=274−6 f(−32)=34 Now substitute x=−32 in g(x)=x2+3x+2 as follows:g(x)=x2+3x+2 g(−32)=(−32)2+3(−32)+2 g(−32)=94−92+2 g(−32)=9−18+84g(−32)=−14Finally substituting x=−32 in the polynomial p(x)=f(x)+3g(x) we get,p(x)=f(x)+3g(x) p(−32)=f(−32)+3g(−32)p(−32)=34+3(−14)p(−32)=34−34 p(−32)=0Hence, the polynomial p(x)=f(x)+3g(x) is divisible by 2x+3.

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