4. explain how you can simplify the expression $\frac{6x^{3}-6x^{2}}{x^{4}+5x^{3}}div\frac{3x^{2}-15x +…

Explanation:

Step1: Factor each polynomial

$6x^{3}-6x^{2}=6x^{2}(x - 1)$; $x^{4}+5x^{3}=x^{3}(x + 5)$; $3x^{2}-15x + 12=3(x^{2}-5x + 4)=3(x - 1)(x - 4)$; $2x^{2}+2x-40=2(x^{2}+x - 20)=2(x + 5)(x - 4)$; $2x^{2}-10x + 12=2(x^{2}-5x + 6)=2(x - 2)(x - 3)$; $4x^{3}+16x^{2}-20x=4x(x^{2}+4x - 5)=4x(x + 5)(x - 1)$.

Step2: Rewrite the original expression with factored - forms

The original expression $\frac{6x^{3}-6x^{2}}{x^{4}+5x^{3}}\div\frac{3x^{2}-15x + 12}{2x^{2}+2x - 40}\times\frac{2x^{2}-10x + 12}{4x^{3}+16x^{2}-20x}$ becomes $\frac{6x^{2}(x - 1)}{x^{3}(x + 5)}\times\frac{2(x + 5)(x - 4)}{3(x - 1)(x - 4)}\times\frac{2(x - 2)(x - 3)}{4x(x + 5)(x - 1)}$.

Step3: Simplify the expression

Cancel out the common factors: $\frac{6x^{2}(x - 1)\times2(x + 5)(x - 4)\times2(x - 2)(x - 3)}{x^{3}(x + 5)\times3(x - 1)(x - 4)\times4x(x + 5)(x - 1)}=\frac{6\times2\times2}{3\times4}\times\frac{x^{2}}{x^{3}\times x}\times\frac{(x - 1)}{(x - 1)\times(x - 1)}\times\frac{(x + 5)}{(x + 5)\times(x + 5)}\times\frac{(x - 4)}{(x - 4)}\times(x - 2)(x - 3)$ $= 1\times\frac{1}{x^{2}}\times\frac{1}{x - 1}\times\frac{1}{x + 5}\times1\times(x - 2)(x - 3)=\frac{(x - 2)(x - 3)}{x^{2}(x - 1)(x + 5)}=\frac{x^{2}-5x + 6}{x^{4}+4x^{3}-5x^{2}}$.

Step4: Find the restrictions on the variable

The original rational - expressions are undefined when the denominators of the original fractions are zero. For $x^{4}+5x^{3}=x^{3}(x + 5)=0$, $x = 0$ or $x=-5$; For $2x^{2}+2x - 40=2(x + 5)(x - 4)=0$, $x=-5$ or $x = 4$; For $4x^{3}+16x^{2}-20x=4x(x + 5)(x - 1)=0$, $x = 0$ or $x=-5$ or $x = 1$. So the restrictions on the variable are $x\neq0,x\neq1,x\neq - 5,x\neq4$.

Answer:

a) The simplified expression is $\frac{x^{2}-5x + 6}{x^{4}+4x^{3}-5x^{2}}$; b) The restrictions on the variable are $x\neq0,x\neq1,x\neq - 5,x\neq4$.