document.write( "Question 389636: For the function f(x)= (2x^3+3x-5)/(x^2+x-30)\r
\n" ); document.write( "\n" ); document.write( "a) The slant asymptote is __________\r
\n" ); document.write( "\n" ); document.write( "b) The vertical asymptotes are __________ \n" ); document.write( "
Algebra.Com's Answer #276176 by haileytucki(390)\"\" \"About 
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f(x)=(2x^(3)+3x-5)/(x^(2)+x-30)\r
\n" ); document.write( "\n" ); document.write( "Factor the polynomial using the rational roots theorem.
\n" ); document.write( "f(x)=((x-1)(2x^(2)+2x+5))/(x^(2)+x-30)\r
\n" ); document.write( "\n" ); document.write( "In this problem 6*-5=-30 and 6-5=1, so insert 6 as the right hand term of one factor and -5 as the right-hand term of the other factor.
\n" ); document.write( "f(x)=((x-1)(2x^(2)+2x+5))/((x+6)(x-5))\r
\n" ); document.write( "\n" ); document.write( "The domain of an expression is all real numbers except for the regions where the expression is undefined. This can occur where the denominator equals 0, a square root is less than 0, or a logarithm is less than or equal to 0. All of these are undefined and therefore are not part of the domain.
\n" ); document.write( "(x+6)(x-5)=0\r
\n" ); document.write( "\n" ); document.write( "Solve the equation to find where the original expression is undefined.
\n" ); document.write( "x=-6,5\r
\n" ); document.write( "\n" ); document.write( "The domain of the rational expression is all real numbers except where the expression is undefined.
\n" ); document.write( "x$-6,x$5_(-I,-6) U (-6,5) U (5,I)\r
\n" ); document.write( "\n" ); document.write( "The vertical asymptotes are the values of x that are undefined in the function.
\n" ); document.write( "x=-6_x=5\r
\n" ); document.write( "\n" ); document.write( "A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches I.
\n" ); document.write( "L[x:I,((x-1)(2x^(2)+2x+5))/((x+6)(x-5))]\r
\n" ); document.write( "\n" ); document.write( "The value of L[x:I,((x-1)(2x^(2)+2x+5))/((x+6)(x-5))] is I.
\n" ); document.write( "I\r
\n" ); document.write( "\n" ); document.write( "There are no horizontal asymptotes because the limit does not exist.
\n" ); document.write( "No horizontal asymptote approaching I.\r
\n" ); document.write( "\n" ); document.write( "A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches -I.
\n" ); document.write( "L[x:-I,((x-1)(2x^(2)+2x+5))/((x+6)(x-5))]\r
\n" ); document.write( "\n" ); document.write( "The value of L[x:-I,((x-1)(2x^(2)+2x+5))/((x+6)(x-5))] is -I.
\n" ); document.write( "-I\r
\n" ); document.write( "\n" ); document.write( "The horizontal asymptote is the value of y as x approaches -I.
\n" ); document.write( "y=-I\r
\n" ); document.write( "\n" ); document.write( "Complete the polynomial division of the expression to determine if there is any remainder.
\n" ); document.write( "x^(2)+x-30,2x^(3)+0x^(2)+3x-5,-2x^(3)-2x^(2)+60x,M2x^(3)-2x^(2)+63x-5,M2x^(3)-2x^(2)+2x-60,M2x^(3)-2x^(2)-65x-65,2x-2\r
\n" ); document.write( "\n" ); document.write( "Split the solution into the polynomial portion and the remainder.
\n" ); document.write( "2x-2+(65x-65)/(x^(2)+x-30)\r
\n" ); document.write( "\n" ); document.write( "The oblique asymptote is the polynomial portion of the long division result.
\n" ); document.write( "y=2x-2\r
\n" ); document.write( "\n" ); document.write( "This is the set of all asymptotes for f(x)=((x-1)(2x^(2)+2x+5))/((x+6)(x-5)).
\n" ); document.write( "Vertical Asymptote: x=-6,x=5_Horizontal Aysmptote:y=-Infinite_Oblique Aysmptote:y=2x-2 \n" ); document.write( "
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