2x^3 - 5x^2 - 11x - 4 = 0
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The Rational Root Theorem tells you
that if the polynomial has a rational
zero then it must be a fraction p/q ,
where p is a factor of the trailing
constant and q is a factor of the
leading coefficient .
The factors of the leading coefficient
(2) are 1 2 .The factors of the constant
term (-4) are 1 2 2 4 . Then the
Rational Roots Tests yields the
following possible solutions:
±1/1 ±1/2 ±2/1 ±2/2 ±2/1 ±2/2 ±4/1
±4/2 . After simplifying we have:
±1 ±2 ±4 ±1/2 ;
putting them one by one, we find p(-1)=0
ie x+1 is one factor of the equation.
dividing the equation by x+1, we get
2x^2 -7x-4
= 2x^2 -8x+x-4
= 2x ( x-4 ) +1 ( x-4 )
= ( 2x+1 ) ( x-4 )
or x= -1/2 or x= 4
ie roots of equation 2x^3 -5x^2 -11x-4 =0 is one are -1 , 4 and -1/2
good luck !
that if the polynomial has a rational
zero then it must be a fraction p/q ,
where p is a factor of the trailing
constant and q is a factor of the
leading coefficient .
The factors of the leading coefficient
(2) are 1 2 .The factors of the constant
term (-4) are 1 2 2 4 . Then the
Rational Roots Tests yields the
following possible solutions:
±1/1 ±1/2 ±2/1 ±2/2 ±2/1 ±2/2 ±4/1
±4/2 . After simplifying we have:
±1 ±2 ±4 ±1/2 ;
putting them one by one, we find p(-1)=0
ie x+1 is one factor of the equation.
dividing the equation by x+1, we get
2x^2 -7x-4
= 2x^2 -8x+x-4
= 2x ( x-4 ) +1 ( x-4 )
= ( 2x+1 ) ( x-4 )
or x= -1/2 or x= 4
ie roots of equation 2x^3 -5x^2 -11x-4 =0 is one are -1 , 4 and -1/2
good luck !
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By inspection, -1 is a root to this equation (11 - 2 - 5 - 4 = 0), so x + 1 is a linear factor. By polynomial division, this factorises to
(x + 1)(2x^2 - 7x - 4)
we notice intuitively that 2*-4 = -8, and -8 + 1 = -7 which is the co-efficient of the middle term, hence we can factorise further: (2x + 1)(x - 4)
(x + 1)(2x + 1)(x - 4) = 0
x = -1, x = -1/2, x = 4.
(x + 1)(2x^2 - 7x - 4)
we notice intuitively that 2*-4 = -8, and -8 + 1 = -7 which is the co-efficient of the middle term, hence we can factorise further: (2x + 1)(x - 4)
(x + 1)(2x + 1)(x - 4) = 0
x = -1, x = -1/2, x = 4.
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OK:
first check -1, 0, and 1. Of these -1 gives us a zero so x+1 is the first. Next use division to find the next few zeroes. so (2x^3-5x^2-11x-4)/(x+1)=2x^2-7x-4
now you can factor to find the last two zeroes. So (2x+1)(x-4) gives us the zeroes of x=4 and -1/2
first check -1, 0, and 1. Of these -1 gives us a zero so x+1 is the first. Next use division to find the next few zeroes. so (2x^3-5x^2-11x-4)/(x+1)=2x^2-7x-4
now you can factor to find the last two zeroes. So (2x+1)(x-4) gives us the zeroes of x=4 and -1/2