document.write( "Question 175797This question is from textbook
\n" ); document.write( ": According to Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots?\r
\n" ); document.write( "\n" ); document.write( "f(a)= a^5-4a^2-7\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "f(x) = 3x^3 + 9x^2 + 8x \n" ); document.write( "
Algebra.Com's Answer #131087 by gonzo(654)\"\" \"About 
You can put this solution on YOUR website!
maximum number of positive roots is number of times the sign of the coefficient changes for f(x).
\n" ); document.write( "---
\n" ); document.write( "maximum number of negative roots is number of times the sign of the coefficient changes for f(-x).
\n" ); document.write( "---
\n" ); document.write( "possible number of roots is:
\n" ); document.write( "number of maximum roots minus 2 until you get below 0.
\n" ); document.write( "example 1:
\n" ); document.write( "maximum number of roots is 6.
\n" ); document.write( "possible number of roots is 6,4,2,0
\n" ); document.write( "---
\n" ); document.write( "example 2:
\n" ); document.write( "maximum number of roots is 7.
\n" ); document.write( "possible number of roots is 7,5,3,1
\n" ); document.write( "note that getting 0 roots in this case is not a possibility.
\n" ); document.write( "---
\n" ); document.write( "if i let x = a, your first equation becomes:
\n" ); document.write( "f(x) = x^5 - 4x^2 - 7
\n" ); document.write( "i could have left the variable is a, but it graphs better when i call it x since some graphing calculators or online graphing calculators only recognize x as the variable.
\n" ); document.write( "---
\n" ); document.write( "your first equation is:
\n" ); document.write( "f(x) = x^5 - 4x^2 - 7
\n" ); document.write( "---
\n" ); document.write( "equation must be in standard form meaning that the degree of the exponents goes from higher to lower as you go from left to right.
\n" ); document.write( "this equation is already in that form as is your second equation so i won't bother to mention this again.
\n" ); document.write( "---
\n" ); document.write( "signs of the coefficients in the first equation are:
\n" ); document.write( "\"+ - -\"
\n" ); document.write( "number of changes in the sign of the coefficients is 1.
\n" ); document.write( "maximum number of positive roots is 1.
\n" ); document.write( "possible number of positive roots is 1.
\n" ); document.write( "---
\n" ); document.write( "to get negative roots, we need to get f(-x).
\n" ); document.write( "f(-x) = (-x)^5 - 4(-x)^2 - 7
\n" ); document.write( "this becomes:
\n" ); document.write( "f(-x) = -x^5 -4x^2 - 7
\n" ); document.write( "signs of the coefficients are:
\n" ); document.write( "\"- - -\"
\n" ); document.write( "maximum number of negative roots is 0.
\n" ); document.write( "possible number of negative roots is 0.
\n" ); document.write( "---
\n" ); document.write( "graph of this equation is:
\n" ); document.write( "\"graph%28600%2C100%2C-10%2C10%2C-10%2C10%2Cx%5E5-4x%5E2-7%29\"
\n" ); document.write( "---
\n" ); document.write( "your second equation is:
\n" ); document.write( "f(x) = 3x^3 + 9x^2 + 8x
\n" ); document.write( "it is in standard form.
\n" ); document.write( "signs of the coefficients are:
\n" ); document.write( "\"+ + +\"
\n" ); document.write( "number of sign changes is 0.
\n" ); document.write( "maximum number of positive roots is 0.
\n" ); document.write( "possible number of positive roots is 0.
\n" ); document.write( "---
\n" ); document.write( "f(-x) = 3*(-x)^3 + 9*(-x)^2 + 8*(-x)
\n" ); document.write( "this is the same as:
\n" ); document.write( "-3x^3 + 9x^2 - 8x
\n" ); document.write( "signs of the coefficients are:
\n" ); document.write( "\"- + -\"
\n" ); document.write( "number of sign changes is 2.
\n" ); document.write( "maximum number of negative roots is 2.
\n" ); document.write( "possible number of negative roots is 2,0.
\n" ); document.write( "---
\n" ); document.write( "graph of this equation is:
\n" ); document.write( "\"graph%28600%2C100%2C-10%2C10%2C-10%2C10%2C3x%5E3+%2B+9x%5E2+%2B+8x%29\"
\n" ); document.write( "this particular equation has 0 negative roots and 0 positive roots.
\n" ); document.write( "it does have a root at x = 0.
\n" ); document.write( "this can be determined from the equation because x factors out and we are left with:
\n" ); document.write( "x * (3x^2 + 9x + 8) = f(x) = 0
\n" ); document.write( "---
\n" ); document.write( "even though the maximum number of negative roots is 2, this equation had 0.
\n" ); document.write( "0 was one of the possible number of roots so descartes rule of signs is accurate.\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );