Simplify and State the restrictions if any. http://postimage.org/image/h1ckmc4un/Rational Expressions

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a) You should simplify the expression `((x^2+6x+5)(x^2+2x-8))/((x^2+7x+12)(x^2-25))/(x^2-x-2)/(x^2-2x-15).`

You should write each term in factored form, hence, you need to find the solutions of the terms such that:

`x^2+6x+5 = 0 `

`x_(1,2) = (-6+-sqrt(36 - 20))/2 `

`x_(1,2) = (-6+-sqrt16)/2`

`x_(1,2) = (-6+-4)/2 `

`x_1 = -1 ; x_2 = -5`

You may write the factored form such that:

`x^2+6x+5 = (x + 1)(x+5)` 

`x^2+2x-8 = 0`

`x_(1,2) = (-2+-sqrt(4+32))/2 `

`x_(1,2) = (-2+-sqrt36)/2`

`x_(1,2) = (-2+-6)/2`

`x_1 = 2 ; x_2 = -4`

`x^2+2x-8 = (x - 2)(x + 4)`

`x^2+7x+12 = 0`

`x_(1,2) = (-7+-sqrt(49-48))/2`

`x_(1,2) = (-7+-1)/2`

`x_1 = -3 ; x_2 = -4`

`x^2+7x+12 = (x+3)(x+4)`

`x^2-25 = (x-5)(x+5)`

`x^2-x-2 = 0`

`x_(1,2) = (1+-sqrt(1+8))/2 `

`x_(1,2) = (1+-3)/2`

`x_1 = 2 ; x_2 = -1 `

`x^2-x-2 = (x-2)(x+1)`

`x^2-2x-15 = 0 `

`x_(1,2) = (2+-sqrt(4+60))/2`

`x_(1,2) = (2+-sqrt64)/2 `

`x_(1,2) = (2+-8)/2`

`x_1 = 5 ; x_2 = -3 `

`x^2-2x-15 = (x-5)(x+3)`

You may write the fraction using the factored forms of the terms such that:

`((x^2+6x+5)(x^2+2x-8))/((x^2+7x+12)(x^2-25))/(x^2-x-2)/(x^2-2x-15) = ((x + 1)(x + 5)(x - 2)(x + 4))/((x+3)(x+4)(x-5)(x+5))/((x-2)(x+1))/((x-2)(x+1))`

Reducing like terms yields:

`((x^2+6x+5)(x^2+2x-8))/((x^2+7x+12)(x^2-25))/(x^2-x-2)/(x^2-2x-15) = ((x - 2)/((x+3)(x-5)))/1/((x+1))`

`((x^2+6x+5)(x^2+2x-8))/((x^2+7x+12)(x^2-25))/(x^2-x-2)/(x^2-2x-15) = ((x - 2)(x+1))/((x+3)(x-5))`

Hence, evaluating the simplifications yields `((x - 2)(x+1))/((x+3)(x-5)).`

b) You should write each term in factored form, hence, you need to find the solutions of the terms such that:

`2x^2 - 7x + 6 = 0`

`x_(1,2) = (7+-sqrt(49 - 48))/4`

`x_(1,2) = (7+-1)/4`

`x_1 = 2 ; x_2 = 3/2`

`2x^2 - 7x + 6 = (x - 2)(x - 3/2)`

`2x^2 - x- 3= 0`

`x_(1,2) = (1+-sqrt(1+24))/4`

`x_(1,2) = (1+-sqrt25)/4`

`x_(1,2) = (1+-5)/4`

`x_1 = 3/2 ; x_2 = -1`

`2x^2 - x - 3 = (x + 1)(x - 3/2)`

You may write the terms in factored form such that:

`(x+1)/((x - 2)(x - 3/2)) - (x-3)/((x + 1)(x - 3/2))`

You need to bring the fractions to a common denominator such that:

`((x+1)^2 - (x-2)(x-3))/((x - 2)(x+1)(x - 3/2))`

`(x^2 + 2x + 1 - x^2 + 5x - 6)/((x - 2)(x+1)(x - 3/2))`

`(7x - 5)/((x - 2)(x+1)(x - 3/2))`

Hence, evaluating the simplifications yields `(7x - 5)/((x - 2)(x+1)(x - 3/2)).`

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