question_answer

Divide the given polynomial by the given monomial.

(a)\[\left( 5{{x}^{2}}-6x \right)\div 3x~~\]

(b) \[\left( 3{{y}^{8}}-4{{y}^{6}}+5{{y}^{4}} \right)\div \text{ }{{y}^{4}}\]

(c) \[8({{x}^{3}}{{y}^{2}}{{z}^{2}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{x}^{2}}{{y}^{2}}{{z}^{3}})\div 4{{x}^{2}}{{y}^{2}}{{z}^{2}}\]

(d) \[({{x}^{3}}+2{{x}^{2}}+\text{ }3x)\div 2x\]

Answer:

(a)   \[(5{{x}^{2}}-6x)\div 3x=\frac{5{{x}^{2}}-6x}{3x}\]

\[=\frac{5{{x}^{2}}}{3x}-\frac{6x}{3x}\]

\[=\frac{5}{3}x-2\]

(b)   \[(3{{y}^{8}}-4{{y}^{6}}+5{{y}^{4}})\div {{y}^{4}}-\frac{3{{y}^{8}}}{{{y}^{4}}}-\frac{4{{y}^{6}}}{{{y}^{4}}}+\frac{5{{y}^{4}}}{{{y}^{4}}}\]

\[=3{{y}^{4}}-4{{y}^{2}}+5\]

(c) \[8({{x}^{3}}{{y}^{2}}{{z}^{2}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{x}^{2}}{{y}^{2}}{{z}^{3}})\div 4{{x}^{2}}{{y}^{2}}{{z}^{2}}\]

\[=\frac{8({{x}^{3}}{{y}^{2}}{{z}^{2}}+{{x}^{2}}{{y}^{3}}{{z}^{2}}+{{x}^{2}}{{y}^{2}}{{z}^{3}})}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}\]

\[=\frac{8{{x}^{3}}{{y}^{2}}{{z}^{2}}}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}+\frac{8{{x}^{2}}{{y}^{3}}{{z}^{2}}}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}+\frac{8{{x}^{2}}{{y}^{2}}{{z}^{3}}}{4{{x}^{2}}{{y}^{2}}{{z}^{2}}}\]

\[=2x+2y+2z=2(x+y+z)\]

(d) \[({{x}^{3}}+2{{x}^{2}}+3x)\div 2x=\frac{{{x}^{3}}+2{{x}^{2}}+3x}{2x}\]

\[=\frac{{{x}^{3}}}{2x}+\frac{2{{x}^{2}}}{2x}+\frac{3x}{2x}\]

\[=\frac{{{x}^{2}}}{2}+x+\frac{3}{2}\]

\[=\frac{1}{2}({{x}^{2}}+2x+3)\]