Give the factors for the numerator [n] and denominator [d] after reducing.

Mathematics

1 answer:

Burka [1]3 years ago

8 0

Answer:

The factors for the numerator [n] and denominator [d] after reducing will be:

$\frac{x^2-25}{x^2-4x}\div \:\frac{2x^2+2x-40}{x^3-x}=\frac{\left(x-5\right)\left(x+1\right)\left(x-1\right)}{2\left(x-4\right)^2}$

Step-by-step explanation:

Given the expression

$\frac{x^2-25}{x^2-4x}\div \frac{2x^2+2x-40}{x^3-x}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$

$=\frac{x^2-25}{x^2-4x}\times \frac{x^3-x}{2x^2+2x-40}$

$\mathrm{Multiply\:fractions}:\quad \frac{a}{b}\times \frac{c}{d}=\frac{a\:\times \:c}{b\:\times \:d}$

$=\frac{\left(x^2-25\right)\left(x^3-x\right)}{\left(x^2-4x\right)\left(2x^2+2x-40\right)}$

$=\frac{\left(x^2-25\right)x\left(x^2-1\right)}{\left(x^2-4x\right)\left(2x^2+2x-40\right)}$

$\mathrm{Cancel\:the\:common\:factor:}\:x$

$=\frac{\left(x^2-25\right)\left(x^2-1\right)}{2\left(x-4\right)\left(x^2+x-20\right)}$

∵ As factor $\left(x^2-25\right)\left(x^2-1\right)=\left(x+5\right)\left(x-5\right)\left(x+1\right)\left(x-1\right)$

so the expression becomes

$=\frac{\left(x+5\right)\left(x-5\right)\left(x+1\right)\left(x-1\right)}{2\left(x-4\right)\left(x^2+x-20\right)}$

∵ As factor $2\left(x-4\right)\left(x^2+x-20\right)=2\left(x-4\right)^2\left(x+5\right)$

so the expression becomes

$=\frac{\left(x+5\right)\left(x-5\right)\left(x+1\right)\left(x-1\right)}{2\left(x-4\right)^2\left(x+5\right)}$

$\mathrm{Cancel\:the\:common\:factor:}\:x+5$

$=\frac{\left(x-5\right)\left(x+1\right)\left(x-1\right)}{2\left(x-4\right)^2}$

Therefore, the factors for the numerator [n] and denominator [d] after reducing will be:

$\frac{x^2-25}{x^2-4x}\div \:\frac{2x^2+2x-40}{x^3-x}=\frac{\left(x-5\right)\left(x+1\right)\left(x-1\right)}{2\left(x-4\right)^2}$

You might be interested in

Daniel is printing out copies of a presentation. It takes 3 minutes to print a color copy and 1 minute to print a black-and-whit

Effectus [21]

Work the information to set inequalities that represent each condition or restriction.

2) Name the variables.

c: number of color copies
b: number of black-and-white copies

3) Model each restriction:

i) It takes 3 minutes to print a color copy and 1 minute to print a black-and-white copy.

3c + b


ii) He needs to print at least 6 copies ⇒ c + b ≥ 6


iv) And must have the copies completed in no more than 12 minutes ⇒

3c + b ≤ 12


4) Additional restrictions are c ≥ 0, and b ≥ 0 (i.e. only positive values for the number of each kind of copies are acceptable)

5) This is how you graph that:

i) 3c + b ≤ 12: draw the line 3c + b = 12 and shade the region up and to the right of the line.

ii) c + b ≥ 6: draw the line c + b = 6 and shade the region down and to the left of the line.

iii) since c ≥ 0 and b ≥ 0, the region is in the first quadrant.

iv) The final region is the intersection of the above mentioned shaded regions.

v) You can see such graph in the attached figure.