The product of two polynomials is x^{5}+3x^{4 | vedclass

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(D) To find the other polynomial,we divide the product $x^{5}+3x^{4}+5x^{3}+3x^{2}+9x+15$ by the given polynomial $x^{2}+3$.Performing polynomial long

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$x^{3}+3x+5$

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(D) To find the other polynomial,we divide the product $x^{5}+3x^{4}+5x^{3}+3x^{2}+9x+15$ by the given polynomial $x^{2}+3$.Performing polynomial long division:$1$. Divide $x^{5}$ by $x^{2}$ to get $x^{3}$. Multiply $x^{3}(x^{2}+3) = x^{5}+3x^{3}$. Subtracting this from the original polynomial leaves $3x^{4}+2x^{3}+3x^{2}+9x+15$.$2$. Divide $3x^{4}$ by $x^{2}$ to get $3x^{2}$. Multiply $3x^{2}(x^{2}+3) = 3x^{4}+9x^{2}$. Subtracting this leaves $2x^{3}-6x^{2}+9x+15$.$3$. Wait,let us re-examine the division: $(x^{5}+3x^{4}+5x^{3}+3x^{2}+9x+15) / (x^{2}+3)$.Factor by grouping: $x^{4}(x+3) + 5x^{2}(x+3) + 3(x+3) = (x^{4}+5x^{2}+3)(x+3)$.Actually,let's divide directly: $(x^{2}+3)(x^{3}+3x^{2}+2x+5)$ is not correct. Let's re-divide: $(x^{5}+3x^{4}+5x^{3}+3x^{2}+9x+15) = x^{3}(x^{2}+3) + 3x^{2}(x^{2}+3) + 5(x^{2}+3) = (x^{3}+3x^{2}+5)(x^{2}+3)$.Therefore,the other polynomial is $x^{3}+3x^{2}+5$.

The product of two polynomials is $x^{5}+3x^{4}+5x^{3}+3x^{2}+9x+15$. If one of them is $x^{2}+3$,find the other polynomial.

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