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Question 21889: For the polynomial function f(x) = x^4 - x^3 - 2x^2 - 4x - 24
Find all real zeroes (x-intercepts)
Divide all real zeroes out using synthetic division
solve the quadratic equation using the quadratic formula (which should give two complex zeroes)
give complete factorization of f(x)
(I know this seems long - but I am so completely lost - this one question will explain about a half 'o dozen problems I am having with my homework)
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! f(x) = x^4 - x^3 - 2x^2 - 4x - 24 ...TRY ,F(0),F(10,F(-1),F(2)...ETC....WE NOTE THAT F(-2) IS ZERO SO X+2 IS A FACTOR...LET US DIVIDE WITH X+2 USING HORNERS SHORT DIVISION.PLEASE COME BACK IF YOU ARE NOT CONVERSANT WITH THIS DIVISION...
-2....|1....-1....-2....-4....-24...........................................
......|0....-2.....6....-8.....24............................................
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......|1....-3.....4....-12.....0
HENCE QUOTIENT IS
X^3-3X^2+4X-12......WE FIND THAT F(3) IS ZERO...SO X-3 IS A FACTOR..AGAIN DIVIDING
WITH X-3 WE GET
3.....|1....-3.....4....-12.....
......|0....3......0.....12.....
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......|1....0......4.....0
HENCE QUOTIENT IS
X^2+4
=(X)^2-(2I)^2
=(X+2I)(X-2I)
HENCE COMPLETE FACTORISATION IS
f(x) = x^4 - x^3 - 2x^2 - 4x - 24 .=(X+2)(X-3)(X+2I)(X-2I)...THE ROOTS OR X INTERCEPTS ARE ....-2,3,2I AND -2I
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