SOLUTION: simplify the following expression: (x^3+x^2-x-27)/(x^2-4x+3) ((x^5+x^2+16x-4)/(x^4-16))-(x/3)

Question 388884: simplify the following expression:
(x^3+x^2-x-27)/(x^2-4x+3)
((x^5+x^2+16x-4)/(x^4-16))-(x/3)
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
Are these seperate questions?? If so:
(x^(3)+x^(2)-x-27)/(x^(2)-4x+3)
In this problem -1*-3=3 and -1-3=-4, so insert -1 as the right hand term of one factor and -3 as the right-hand term of the other factor.
(x^(3)+x^(2)-x-27)/((x-1)(x-3))

and

((x^(5)+x^(2)+16x-4)/(x^(4)-16))-((x)/(3))
The binomial can be factored using the difference of squares formula, because both terms are perfect squares.
((x^(5)+x^(2)+16x-4)/((x^(2)+4)(x^(2)-4)))-((x)/(3))
The binomial can be factored using the difference of squares formula, because both terms are perfect squares. The difference of squares formula is a^(2)-b^(2)=(a-b)(a+b).
((x^(5)+x^(2)+16x-4)/((x^(2)+4)(x-2)(x+2)))-((x)/(3))
Multiply -1 by the (x)/(3) inside the parentheses.
(x^(5)+x^(2)+16x-4)/((x^(2)+4)(x-2)(x+2))-(x)/(3)
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 3(x^(2)+4)(x+2)(x-2). The ((x^(5)+x^(2)+16x-4))/((x^(2)+4)(x-2)(x+2)) expression needs to be multiplied by ((3))/((3)) to make the denominator 3(x^(2)+4)(x+2)(x-2). The -(x)/(3) expression needs to be multiplied by ((x^(2)+4)(x+2)(x-2))/((x^(2)+4)(x+2)(x-2)) to make the denominator 3(x^(2)+4)(x+2)(x-2).
(x^(5)+x^(2)+16x-4)/((x^(2)+4)(x-2)(x+2))*(3)/(3)-(x)/(3)*((x^(2)+4)(x+2)(x-2))/((x^(2)+4)(x+2)(x-2))
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 3(x^(2)+4)(x+2)(x-2).
((x^(5)+x^(2)+16x-4)(3))/(3(x^(2)+4)(x+2)(x-2))-(x)/(3)*((x^(2)+4)(x+2)(x-2))/((x^(2)+4)(x+2)(x-2))
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 3(x^(2)+4)(x+2)(x-2).
((x^(5)+x^(2)+16x-4)(3))/(3(x^(2)+4)(x+2)(x-2))-(x(x^(2)+4)(x+2)(x-2))/(3(x^(2)+4)(x+2)(x-2))
The numerators of expressions that have equal denominators can be combined. In this case, ((x^(5)+x^(2)+16x-4)(3))/(3(x^(2)+4)(x+2)(x-2)) and -((x(x^(2)+4)(x+2)(x-2)))/(3(x^(2)+4)(x+2)(x-2)) have the same denominator of 3(x^(2)+4)(x+2)(x-2), so the numerators can be combined.
((x^(5)+x^(2)+16x-4)(3)-(x(x^(2)+4)(x+2)(x-2)))/(3(x^(2)+4)(x+2)(x-2))
Simplify the numerator of the expression.
(2x^(5)+3x^(2)+64x-12)/(3(x^(2)+4)(x+2)(x-2))
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