obtain all zeros of polynomial p(x) =2x^4+x^3-14x^2-19x-6 if two of its zeros are -2 and -1
Answers
Answer:
-1/2 and +3
Step-by-step explanation:
if roots of quadratic equation are -2 and -1 then equation
=x²-(α+β)x+αβ
=x²-(-3)x+2
=x²+3x+2
by division lemma
divisor=dividend x quotient + remainder
i.e. 2x⁴+x³-14x²-19x-6=(x²+3x+2)q(x)+r(x)
x²+3x+2 ) 2x⁴+x³-14x²-19x-6 ( 2x²-5x-3
2x⁴+6x³+4x²
( - ) ( - ) ( - )
-5x³-18x²-19x
-5x³-15x²-10x
( + ) ( + ) ( + )
-3x²-9x-6
-3x²-9x-6
( + ) ( + ) ( + )
0
---------------------
q(x)= 2x²-5x-3
=2x²-6x+x-3
=2x(x-3)+1(x-3)
= (2x+1)(x-3)
therefore, roots are -1/2 and +3
Answer:
polynomial has total of 4 roots
let these are a,b,c,d
now applying properties of these roots
a+b+c+d=-1/2
c+d=5/2. ........1
ab+bc+cd+da=-14/2=-7
-c+cd-2d=-9. .........2
abcd=-3
cd=3÷2=1.5. -c-2d=-9-1.5
c+2d=10.5
on solving 1 and 2 we get c=1.5÷8 and d=8
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