document.write( "Question 804507: The polynomial g(x)= -3x^3-15x^2-6x+24 has a zero at x=-2. Find all the other zeros of g(x) using polynomial long division. \n" ); document.write( "
Algebra.Com's Answer #484794 by DrBeeee(684)\"\" \"About 
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Given:
\n" ); document.write( "(1) g(x) = -3x^3 -15x^2 -6x +24
\n" ); document.write( "We are told that a root (zero) of this trinomial is
\n" ); document.write( "(2) x = -2
\n" ); document.write( "Therefore
\n" ); document.write( "(3) x + 2 = 0
\n" ); document.write( "which makes (x + 2) a factor of g(x). That is, if you divide g(x) by (x + 2) you will obtain a quadratic
\n" ); document.write( "(4) ax^2 +bx +c
\n" ); document.write( "Long division is the easy way to do the division which I'm sure you've had in class, but I can't type it here. It's just like dividing numbers. I will you show you how to get the answer using multiplication, ok?
\n" ); document.write( "Before we begin, remember we are deriving the zeroes of g(x), which means we will set g(x) = 0. If we set (1) equal to zero we get
\n" ); document.write( "(5) -3x^3 -15x^2 -6x +24 = 0
\n" ); document.write( "Note that a common factor of all terms is -3, so we have
\n" ); document.write( "(6) -3*(x^3 +5x^2 +2x -8) = 0 or equivalently
\n" ); document.write( "(7) x^3 +5x^2 +2x -8 = 0, much friendlier!
\n" ); document.write( "We know that (7) factors into
\n" ); document.write( "(8) (x+2)*(ax^2+bx+c) = x^3+5x^2+2x-8
\n" ); document.write( "As stated above (and by the question) we can find a,b, and c by dividing both sides of (8) by the known factor (x+2), but I can't do that here.
\n" ); document.write( "Alternatively, multiply out the left side of (8) to get
\n" ); document.write( "(9) a*x^3+b*x^2+c*x+2ax^2+2bx+2*c = x^3+5x^2+2x-8
\n" ); document.write( "Since the left side equals the right side of (9), equate like terms to get
\n" ); document.write( "(10) a*x^3 = x^3
\n" ); document.write( "(11) (b + 2*a)*x^2 = 5*x^2
\n" ); document.write( "(12) (c + 2*b)*x = 2*x and
\n" ); document.write( "(13) 2*c = -8
\n" ); document.write( "We can solve (10) for a
\n" ); document.write( "(14) a = 1
\n" ); document.write( "and (13) for c
\n" ); document.write( "(15) c = -4
\n" ); document.write( "In (11), set the coefficients equal to get
\n" ); document.write( "(16) b + 2*a = 5 and use a = 1 to get
\n" ); document.write( "(16) b + 2*1 = 5 or
\n" ); document.write( "(17) b = 5 - 2 or
\n" ); document.write( "(18) b = 3
\n" ); document.write( "We don't need (12), but we can verify a,b,and c.
\n" ); document.write( "In (12), set the coefficients equal to get
\n" ); document.write( "(19) c + 2*b = 2 and use c = -4 and b = 3 to get
\n" ); document.write( "Is (-4 + 2*3 = 2)?
\n" ); document.write( "Is (-4 + 6 = 2)?
\n" ); document.write( "Is (2 = 2)? Yes
\n" ); document.write( "The quadratic factor of (4) is
\n" ); document.write( "(20) x^2 +3*x -4 or
\n" ); document.write( "(21) g(x) = -3*(x+2)*(x^2+3x-4)
\n" ); document.write( "Now we can further factor the quadratic of (20) to get
\n" ); document.write( "(22) x^2 +3*x -4 = (x+4)*(x-1)
\n" ); document.write( "which when set equal to zero gives
\n" ); document.write( "(23) x + 4 = 0 or
\n" ); document.write( "(24) x = -4
\n" ); document.write( "and
\n" ); document.write( "(25) x -1 = 0 or
\n" ); document.write( "(26) x = 1
\n" ); document.write( "Answer: The zeroes of g(x) are {-4,-2,1}\r
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