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2. Polynomials
hard
जाँच कीजिए कि $x+2$ बहुपदों $x^{3}+3 x^{2}+5 x+6$ और $2 x+4$ का एक गुणनखंड है या नहीं।
Option A
Option B
Option C
Option D
Solution
The zero of $x+2$ is $-2$ . Let $p(x)=x^{3}+3 x^{2}+5 x+6$ and $s(x)=2 x+4$
Then, $p(-2)=(-2)^{3}+3(-2)^{2}+5(-2)+6$
$=-8+12-10+6$
$=0$
So, by the Factor Theorem, $x+2$ is a factor of $x^{3}+3 x^{2}+5 x+6$
Again, $s(-2)=2(-2)+4=0$
So, $x+2$ is a factor of $2 x+4 .$ In fact, you can check this without applying the Factor Theorem, since $2 x+4=2(x+2)$.
Standard 9
Mathematics