Find the zeros of the polynomial 4x4-20x3+23x2+5x-6 if two of the zeros are 3 and 2
Answers
Given that 2, 3 are the zeros of given polynomial 4x*4-20x*3+23x*2+5x-6
So, x = 2 x = 3
x-2=0 x-3=0
So, (x-2)(x-3) is the factor of given polynomial 4x*4-20x*3+23x*2+5x-6.
x²-5x+6 is the factor of given polynomial 4x*4-20x*3+23x*2+5x-6.
On dividing 4x*4-20x*3+23x*2+5x-6 with x²-5x+6
x²-5x+6) 4x*4-20x³+23x²+5x-6 (4x²-1 --- Quotient
4x*4-20x³+24x²
(-) (+) (-)
___________________
0 0 -x²+5x-6
-x²+5x-6
(+) (-) (+)
___________________
0 Remainder
____________________
Here quotient is 4x²-1. So, it must be a factor of given polynomial 4x*4-20x*3+23x*2+5x-6
So, 4x²-1=0
4x² = 1
x² = 1/4
x = √1/4
x = + or - 1/2
x = 1/2 or -1/2
Therefore other roots of given polynomial 4x*4-20x*3+23x*2+5x-6 are 1/2, -1/2
There’s an alternate method too:
2 and 3 are the zero of polynomial.
then x= 2 also x= 3
(x-2) (x-3) is a factor of this polynomial
(x-2) (x-3)= x^2-5x+6.
Divide the polynomial by the factor we get : 4x^4 - 20x^3 +23x^2 +5x-6 / x^2 -5x +6.
Q(x) = 4x^2 - 1 .
Factorize 4x^2 - 1 we get : (2x ) ^2 - (1)^2 .
= (2x+1) (2x-1).
= 2x+1 =0 , x= -1/2 .
= 2x - 1 =0 , x = 1/2 .
it's other two zero are -1/2 , 1/2 .
Hope This Helps :)
Answer:
p(x) = 4x^4 - 20x^3 + 23x^2 + 5x - 6
2 and 3 are zeroes of p(x)
alpha=2
beta=3
alpha+beta= 2+3 = 5
alpha*beta=2*3= 6
x^2-(alpha + beta)x + alpha*beta <------ formula
x^2-5x+6
divide 4x^4 - 20x^3 + 23x^2 + 5x - 6 by x^2-5x+6
x²-5x+6 ) 4x*4-20x³+23x²+5x-6 ( 4x²-1
4x*4-20x³+24x²
-----------------------
-x²+5x-6
-x²+5x-6
-------------------
0
splitting the middle term of 4x²-1
4x² = 1
x² = 1 /4
x² = 1/2 and -1/2
the zeroes other than 2 and 3 are 1/2 and -1/2