Question #6641c

1 Answer
Jim G.
Jan 20, 2018

#(x-1)(x-4)(x+2)(2x-3)#

Explanation:

#x^2-5x+4=(x-1)(x-4)#

#"since "(x-1)" and "(x-4)" are factors"#

#"then "x=1" and "x=4" are roots of the polynomial"#

#"substitute each of these values into the polynomial"#

#x=1to2+h+k+34-24=0#

#x=4to512+64h+16k+136-24=0#

#rArrh+k=-12to(1)#

#rArr64h+16k=-624to(2)#

#"from equation "(1)toh=-12-kto(3)#

#"substitute "h=-12-k" into equation "(2)#

#64(-12-k)+16k=-624#

#rArr-768-64k+16k=-624#

#rArr-48k=144rArrk=-3#

#"substitute "k=-3" into equation "(3)#

#rArrh=-12+3=-9#

#"the polynomial can now be expressed as"#

#2x^4-9x^3-3x^2+34x-24#

#"now divide the polynomial by the factor "x^2-5x+4#

#(2x^4-9x^3-3x^2+34x-24)/(x^2-5x+4)#

#=2x^2+x-6=(x+2)(2x-3)#

#rArr2x^4-9x^3-3x^2+34x-24#

#=(x-1)(x-4)(x+2)(2x-3)#

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